Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11...

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Lecture 11 Moist Processes – Part 2 Unsaturated – Skew T Review Clausius – Clapeyron Equation Saturation Mixing Ratio Moist Adiabatic Lapse Rate Lifting Condensation Level ATMOS 5130

Transcript of Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11...

Page 1: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

Lecture11

• MoistProcesses– Part2• Unsaturated– SkewT• ReviewClausius – Clapeyron Equation• SaturationMixingRatio• MoistAdiabaticLapseRate• LiftingCondensationLevel

ATMOS5130

Page 2: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

SimplestformofSkewT

12Z 25 Aug 2005 Green Bay, WI

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Skew T - log pb)ReviewReview

Page 3: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

StandardMeteorologicalformofSkewT

ReviewGoal:

Understandmeaningofalllines.

Page 4: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

-40 -30 -20 -10 0 10 20 30 40

-40-50-60-70-80-90-100-110-120

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C]

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1000273 K

253 K

233 K293 K

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393 K

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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.

Page 5: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 6: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

SaturationMixingRatio𝑤" 𝑇, 𝑃 = 567(9)

;<67(9)

𝑤" 𝑇, 𝑃 ≈ 567(9);

𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑅B𝑅,

Uniquevalueofws froanycombinationofTandp

Thus,linesmaybedrawnonaskew-T

Page 7: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

SaturationMixingRatio𝑤" 𝑇, 𝑃 = 567(9)

;<67(9)

𝑤" 𝑇, 𝑃 ≈ 567(9);

-40 -30 -20 -10 0 10 20 30 40

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hP

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DashedLine=SaturationMixingRatio

𝑊ℎ𝑒𝑟𝑒: 𝜀 = 𝑅B𝑅,

Page 8: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation
Page 9: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

DewPointandMixingRatio

TTd

p

wsw

Page 10: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

DewPointDepressionandRelativeHumidity100

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Quillayute, WA 2004 9/11 12ZΔ𝑇B = 𝑇 − 𝑇B

WhereT=airtemperatureTd=dewpointtemperature

Page 11: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

DewPointDepressionandRelativeHumidity100

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Quillayute, WA 2004 9/11 12Z

NearlySaturatedHighRHNear100%

Δ𝑇B = 𝑇 − 𝑇B

WhereT=airtemperatureTd=dewpointtemperature

Page 12: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

DewPointDepressionandRelativeHumidity100

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Pre

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re (

hP

a) 2

73 K

253 K

233 K293 K

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2 g/kg

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Quillayute, WA 2004 9/11 12Z

DryLowRH

Δ𝑇B = 𝑇 − 𝑇B

WhereT=airtemperatureTd=dewpointtemperature

Page 13: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

LiftingCondensationLevel(LCL)

• Pressureatwhichsaturationisachievedbyaparcelduringadiabaticascent• Levelatwhichtoexpectacloudbasetoform

• SkewT– Intersectionofthedryadiabatcorrespondingtotheparceltemperatureandthemixingratiolinecorrespondingtotheparcel’sdewpoint

Page 14: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

LiftingCondensationLevel(LCL)- Approximations

𝐿𝐶𝐿 ≈ 𝑝 exp −0.044Δ𝑇B

𝐿𝐶𝐿 𝑘𝑚 ≈ Δ𝑇B /8

WheredewpointdepressionisgivenindegreesC

WheredewpointdepressionisgivenindegreesK

Page 15: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

MoistAdiabaticLapseRate

Alsocalledsaturation-adiabaticlapserateorpseudoadiabaticRecalladiabaticmeansnooutsideheatisinvolvedinwarmingorcoolingofairparcel

Whyisthedryandmoistadiabaticlapseratedifferent?

Watervaporinarisingparcelofairwillcondensewhentheairbecomescoldenough.Thephasechangefromgastoliquidtakesalittleworkfromthewatermolecules.Astheyareworking,theyreleaseheat.Theheatdecreasesthecoolingthatoccursintheairparcel.Therefore,arisingparcelofdryaircoolsfasterthanamoistparcelofair.Conversely,asinkingparcelofdryairwarmsfasterthanasinkingparcelofmoistair.

Page 16: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 17: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

SpecialCasesoftheFirstLaw

𝛿𝑞 = 𝑐,𝑑𝑇 + 𝑝𝑑𝛼𝛿𝑞 = 𝑐;𝑑𝑇 − 𝛼𝑑𝑝

DryAdiabaticProcess:(𝛿q=0)negligiblechangeofheatbetweenthesystemandenvironment

𝑐,𝑑𝑇 = −𝑝𝑑𝛼𝑐;𝑑𝑇 = 𝛼𝑑𝑝

SpecialSignificanceinMeteorology!!

Review

Page 18: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 19: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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;

Page 20: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 21: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 22: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

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

Page 23: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation
Page 24: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

InClassproblem:Problem7.10:

Usingtheskew-Tdiagramatthebackofthebookdeterminethesaturationvaporpressureatatemperatureof– 20C

Page 25: Lecture 11 - University of Utahhome.chpc.utah.edu/~hallar/Thermo/Lectures/Lecture11.pdfLecture 11 • Moist Processes –Part 2 • Unsaturated –Skew T • Review Clausius–ClapeyronEquation

InClassproblem:Problem7.10:

Usingtheskew-Tdiagramdeterminethesaturationvaporpressureatatemperatureof– 20C

Assumedsealevel(1013mb).

~0.78g/kg