Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11...
Transcript of Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11...
Lecture11
• MoistProcesses– Part2• Unsaturated– SkewT• ReviewClausius – Clapeyron Equation• SaturationMixingRatio• MoistAdiabaticLapseRate• LiftingCondensationLevel
ATMOS5130
SimplestformofSkewT
12Z 25 Aug 2005 Green Bay, WI
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hP
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-60 -50
T - log pa)
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Temperature [oC]
Pre
ssu
re (
hP
a)
Skew T - log pb)ReviewReview
StandardMeteorologicalformofSkewT
ReviewGoal:
Understandmeaningofalllines.
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Temperature [o
C]
Pre
ssu
re (
hP
a)
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1000273 K
253 K
233 K293 K
313 K
333 K
353 K
373 K
393 K
413 K
433 K
Fig.5.3
UnsaturatedAtmosphereReview
The word adiabatic means that no outside heat is involved in the warming or cooling of the air parcels.
Clausius – Clapeyron equationfortheatmosphere
𝑒" 𝑇 = 𝑒"& exp𝐿𝑅,
1𝑇&−1𝑇
MeasuredvalueofsaturationvaporpressureatT0T0 =0C=273Keso =6.11hPaL=2.50x106J/kgRv =461.5J/(kgK)
VaporPressurewithrespecttowater
𝑒" 𝑇 ≈ 𝐴 exp−𝐵𝑇
A=2.53x1011 PaB=5420K
SaturationMixingRatio𝑤" 𝑇, 𝑃 = 567(9)
;<67(9)
𝑤" 𝑇, 𝑃 ≈ 567(9);
𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑅B𝑅,
Uniquevalueofws froanycombinationofTandp
Thus,linesmaybedrawnonaskew-T
SaturationMixingRatio𝑤" 𝑇, 𝑃 = 567(9)
;<67(9)
𝑤" 𝑇, 𝑃 ≈ 567(9);
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Temperature [o
C]
Pre
ssu
re (
hP
a)
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253 K
233 K293 K
313 K
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0.2 g/kg 0.4 1
2 g/kg
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DashedLine=SaturationMixingRatio
𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑅B𝑅,
DewPointandMixingRatio
TTd
p
wsw
DewPointDepressionandRelativeHumidity100
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Temperature [o
C]
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ssu
re (
hP
a) 2
73 K
253 K
233 K293 K
313 K
333 K
353 K
373 K
393 K
413 K
433 K
0.2 g/kg 0.4 1
2 g/kg
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Quillayute, WA 2004 9/11 12ZΔ𝑇B = 𝑇 − 𝑇B
WhereT=airtemperatureTd=dewpointtemperature
DewPointDepressionandRelativeHumidity100
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Temperature [o
C]
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ssu
re (
hP
a) 2
73 K
253 K
233 K293 K
313 K
333 K
353 K
373 K
393 K
413 K
433 K
0.2 g/kg 0.4 1
2 g/kg
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Quillayute, WA 2004 9/11 12Z
NearlySaturatedHighRHNear100%
Δ𝑇B = 𝑇 − 𝑇B
WhereT=airtemperatureTd=dewpointtemperature
DewPointDepressionandRelativeHumidity100
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1000
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-40-50-60-70-80-90-100-110-120
Temperature [o
C]
Pre
ssu
re (
hP
a) 2
73 K
253 K
233 K293 K
313 K
333 K
353 K
373 K
393 K
413 K
433 K
0.2 g/kg 0.4 1
2 g/kg
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24
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Quillayute, WA 2004 9/11 12Z
DryLowRH
Δ𝑇B = 𝑇 − 𝑇B
WhereT=airtemperatureTd=dewpointtemperature
LiftingCondensationLevel(LCL)
• Pressureatwhichsaturationisachievedbyaparcelduringadiabaticascent• Levelatwhichtoexpectacloudbasetoform
• SkewT– Intersectionofthedryadiabatcorrespondingtotheparceltemperatureandthemixingratiolinecorrespondingtotheparcel’sdewpoint
LiftingCondensationLevel(LCL)- Approximations
𝐿𝐶𝐿 ≈ 𝑝 exp −0.044Δ𝑇B
𝐿𝐶𝐿 𝑘𝑚 ≈ Δ𝑇B /8
WheredewpointdepressionisgivenindegreesC
WheredewpointdepressionisgivenindegreesK
MoistAdiabaticLapseRate
Alsocalledsaturation-adiabaticlapserateorpseudoadiabaticRecalladiabaticmeansnooutsideheatisinvolvedinwarmingorcoolingofairparcel
Whyisthedryandmoistadiabaticlapseratedifferent?
Watervaporinarisingparcelofairwillcondensewhentheairbecomescoldenough.Thephasechangefromgastoliquidtakesalittleworkfromthewatermolecules.Astheyareworking,theyreleaseheat.Theheatdecreasesthecoolingthatoccursintheairparcel.Therefore,arisingparcelofdryaircoolsfasterthanamoistparcelofair.Conversely,asinkingparcelofdryairwarmsfasterthanasinkingparcelofmoistair.
Clausius – Clapeyron equationfortheatmosphere𝑑𝑒"𝑑𝑇 =
sM − sN𝛼N − 𝛼M
=𝐿
𝑇(𝛼N − 𝛼M)SpecificVolumeofliquidwater
SpecificVolumeofwatervapor
Assumption:𝛼N>>>𝛼M
𝑑𝑒"𝑑𝑇 =
𝐿𝑇(𝛼N)
𝑑𝑒"𝑑𝑇 =
𝐿𝑒"𝑅,𝑇N
Substituteintheidealgaslawforwatervapor1𝛼N
=𝑒"𝑅,𝑇
𝑑𝑒"𝑒"
=𝐿𝑅,
∗𝑑𝑇𝑇N
Q𝑑𝑒"𝑒"
67
67R=𝐿𝑅,Q
𝑑𝑇𝑇N
9
9R𝑒" 𝑇 = 𝑒"& exp
𝐿𝑅,
1𝑇&−1𝑇
Review
SpecialCasesoftheFirstLaw
𝛿𝑞 = 𝑐,𝑑𝑇 + 𝑝𝑑𝛼𝛿𝑞 = 𝑐;𝑑𝑇 − 𝛼𝑑𝑝
DryAdiabaticProcess:(𝛿q=0)negligiblechangeofheatbetweenthesystemandenvironment
𝑐,𝑑𝑇 = −𝑝𝑑𝛼𝑐;𝑑𝑇 = 𝛼𝑑𝑝
SpecialSignificanceinMeteorology!!
Review
MoistAdiabaticLapseRateB9BW= Γ"
𝛿𝑞 = 𝑐;𝑑𝑇 − 𝛼𝑑𝑝 Now: 𝛿𝑞 ≠ 0 asthisrepresentsthelatentheatreleasedtotheairbycondensation.
𝛿𝑞 = −𝐿,𝑑𝑤" Heatgeneratedbycondensationofwater(decreasedvapor)
−𝐿,𝑑𝑤" = 𝑐;𝑑𝑇 − 𝛼𝑑𝑝
−𝐿, 𝜕𝑤"𝜕𝑇 𝑑𝑇 +
𝜕𝑤"𝜕𝑝 𝑑𝑝 = 𝑐;𝑑𝑇 − 𝛼𝑑𝑝
𝛼 − 𝐿,𝜕𝑤"𝜕𝑝 𝑑𝑝 = 𝑐; + 𝐿,
𝜕𝑤"𝜕𝑇 𝑑𝑇
𝛼 − 𝐿,𝜕𝑤"𝜕𝑝
𝑐; + 𝐿,𝜕𝑤"𝜕𝑇
=𝑑𝑝𝑑𝑇
MoistAdiabaticLapseRateB9BW= Γ"
𝛼 −𝐿,𝝏𝒘𝒔𝝏𝒑
𝑐; + 𝐿,𝝏𝒘𝒔𝝏𝑻
=𝑑𝑇dp
𝑤" 𝑇, 𝑃 ≈ 567(9);
RECALL 𝑑𝑒"𝑑𝑇 =
𝐿,𝑒"𝑅,𝑇N
ab7a9
≈ 5;B67B9
= 5;cd67ed9f
= cdb7ed9f
𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑅B𝑅,
𝜕𝑤"𝜕𝑝 ≈ −
𝜀𝑝N 𝑒" = −
1𝑝𝑤"
So, 𝑤"𝜀 𝑝 ≈ 𝑒"
And
ghcd(ijb7)
kjhcdldm7ndof
=gh(ldj b7)
kjhldfm7ndof
=gh(ldpnqo
b7)
kjhldfm7ndof
= gkj
Mh( ldnqob7)
Mh ldfm7rjndof
= B9B;
MoistAdiabaticLapseRate
Invokingthehydrostaticequation
𝑔𝑑𝑧 ≈ −𝛼𝑑𝑝
Γ" ≡ −B9BW=
Mhlm7nqo
Mh lfm7ndrjof
vkj
=Mhlm7nqo
Mh lfm7ndrjof
ΓB
𝛼𝑐;
1 + ( 𝐿,𝑅B𝑇𝑤")
1 + 𝐿,N𝑤"𝑐;𝑅,𝑇N
=𝑑𝑇𝑑𝑝
- gkj
Mh( ldnqob7)
Mh ldfm7rjndof
vg= B9
B;B;BW
MoistAdiabaticLapseRate
𝑑 ln 𝑇𝑑 ln 𝑝 ≈
1 + 𝐿𝑤"𝑅B𝑇
1 + 𝐿N𝑤"𝑅,𝑐;𝑇N
𝑅B𝑐;
Ifws isapproximatelyzero(asisthecaseforverycoldparcels)
ThenthisreducestothedifferentialformofPoisson’sequation.
SONOdistinctionbetweendryandmoistadiabaticascent.
𝛼𝑐;
1 + ( 𝐿,𝑅B𝑇𝑤")
1 + 𝐿,N𝑤"𝑐;𝑅,𝑇N
=𝑑𝑇𝑑𝑝
𝛼 = 𝑅B𝑇𝑝
PotentialTemperature(Poisson’sEquation)
𝑐;𝑑𝑇 = 𝛼𝑑𝑝 = 𝑅B𝑇𝑝 𝑑𝑝
1𝑇 𝑑𝑇 =
𝑅B𝑐;1𝑝 𝑑𝑝
Q1𝑇 𝑑𝑇
9
9R=𝑅B𝑐;Q
1𝑝 𝑑𝑝
;
;R
𝑇𝑇&=
𝑝𝑝&
y
𝜅 = 𝑅B𝑐;≈ 0.286
Review
InClassproblem:Problem7.10:
Usingtheskew-Tdiagramatthebackofthebookdeterminethesaturationvaporpressureatatemperatureof– 20C
InClassproblem:Problem7.10:
Usingtheskew-Tdiagramdeterminethesaturationvaporpressureatatemperatureof– 20C
Assumedsealevel(1013mb).
~0.78g/kg