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Dynamic Meteorology 2016 (lecture 10)

(a.j.vandelden@uu.nl) (http://www.phys.uu.nl/~nvdelden/dynmeteorology.htm)

Topics this week

Quasi-balance in the atmosphere (section 1.30) Structure of balanced vortices: gradient wind balance Thermal wind balance in an axisymmetric vortex Thermal wind balance: potential vorticity inversion equation Solution of the potential vorticity inversion equation What can we learn from the solution of the PV-inversion eq.? “Ageostrophic” effects (section 1.31) Jet streaks (section 1.32)

Gradient wind balance in an axi-symmetric vortex

u>0

r

u<0

Radialwind:

fortheswirlingwind,u

(gradientwindbalance)

Balanceinanaxisymmtericvortexinisentropiccoordinatescanbeexpressedas

Box1.12

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Thermal wind balance in an axi-symmetric vortex

HydrostaCcbalance:

Thermalwindbalance:

Gradientwindbalance:

eliminateΨ

Box1.12

Thermal wind balance in an axi-symmetric vortex

Thermalwindbalance:

Box1.12

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Potential vorticity inversion

f=constant

Previousslide:

(thermalwindbalance)

(PV-inversionequaCon)

Box1.12

Potential vorticity inversion

Anotherwayofexpressingthermalwindbalance:

(PV-inversionequaCon)

PV-inversionequaConisanellipCcparCaldifferenCalequaConif

InthatcasethisequaConhasauniquesoluCon,ifboundarycondiConsarespecified.

YoucanfinduifyouknowZ(andu attheboundariesofthedomainofthesoluCon).

Box1.12

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Boundary conditions

(PV-inversionequaCon)

Whatdoesthischoicemean?

ThestandardformulaConoftheboundarycondiConforthesoluConsofanellipCcsecondorderparCaldifferenCalequaConistospecifyuontheboundary(Dirichletboundarycondi1on)ortospecifythenormalderivaCveofuontheboundary(Neumannboundarycondi1on).Intheproblemathand,weimposeaDirichletboundarycondionatthepole,attheupperboundaryandat10°N.Atthelowerboundary,whichisnotasmoothcurve,weimposeanumericalapproximaConoftheNeumannboundarycondiConbyprescribing∂u/∂θ.

Becauseofthenon-linearityofthePV-inversioneq.andbecauseofthecomplexmixedboundarycondi1ons,thesoluConofthisequaConisfarfromastandardmathemaCcalproblem.

Box1.12

Elliptic equations appear when some kind of balance is assumed

EXAMPLE:Geostrophicbalanceinpressurecoordinates(seeslide5):

f=constant

ThisisPoisson’sequaCon,whichisalsoanellipCcequaCon.

DescribestherelaConbetweenstreamfuncConandrelaCvevorCcityinabalancedatmosphere.

Box1.12

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potential vorticity inversion PV-inversionequaCon:

Thegradientwind(bluecontours,labeledinms-1)asafunctionofradiusandpotentialtemperatureinanatmospherewithanaxisymmetricPV-anomalycentredatθ0=330Kandr=0.ThePV-anomalyhasacharacteristicverticalscale,Δθ=10Kandacharacteristicradialscale,Δr=1000km.ThispanelshowstheresultforZ0=5Zref.BlackcontoursrepresentisoplethsofPVasafractionofthereferencePV.Greencontoursrepresentisoplethsofpotentialvorticity,labeledinPVU.Thethickgreenline(2PVU)representsthedynamicaltropopause.RedsolidcontoursrepresentisobarslabeledinhPa.ThereddashedlinesrepresentisobarsinanatmospherewithoutthePV-anomaly.

Posi<vePV-anomalyf=constant!

NumericalsoluCon(methodisdescribedinchapter7)Box1.12

potential vorticity inversion PV-inversionequaCon:

Thegradientwind(bluecontours,labeledinms-1)asafunctionofradiusandpotentialtemperatureinanatmospherewithanaxisymmetricPV-anomalycentredatθ0=330Kandr=0.ThePV-anomalyhasacharacteristicverticalscale,Δθ=10Kandacharacteristicradialscale,Δr=1000km.ThispanelshowstheresultforZ0=-0.5Zref.BlackcontoursrepresentisoplethsofPVasafractionofthereferencePV.Greencontoursrepresentisoplethsofpotentialvorticity,labeledinPVU.Thethickgreenline(2PVU)representsthedynamicaltropopause.RedsolidcontoursrepresentisobarslabeledinhPa.ThereddashedlinesrepresentisobarsinanatmospherewithoutthePV-anomaly.

Nega<vePV-anomaly

NumericalsoluCon(methodisdescribedinchapter7)Box1.12

f=constant!

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Solution due to Kleinschmidt (1957)

Acyclone“produced”byabodyof6CmesthenormalpotenCalvorCcity.Theundisturbedreferencestateconsistsoftwolayerswithaconstanttemperaturelapserate:atropospherewithdT/dz=5.8°Ckm-1andanisothermalstratosphere.Theleb-handdiagramshowsthetemperatureontheaxis(Ta)andintheundisturbedatmosphere(Tu)asafuncConheight.Cross-secCon:thinlinesareisentropeslabeledinK;heavylinesintheleFhalfindicatetherela<vedepression(pu-p)/pu(perthousand)(puisthepressureintheundisturbedatmosphere).Heavylinesontherightareisotachs,labeledinms-1.

Box1.12

Solution due to Keinschmidt

FromthesoluConofthepotenCalvorCcityinversionequaConweareabletodrawthefollowingthreegeneralconclusions.

1.Withinanisolatedairmasswithabnormalpoten<alvor<citythesta<cstabilityaswellastheabsolutevor<citydeviatefromthenormalinthesamesenseasthepoten<alvor<city.2.In(cyclo-)geostrophicequilibrium,anairmassofrelaCvelyhighpotenCalvorCcityestablishesacyclone.Belowthisairmass,theisentropicsurfacesareraised;abovethisairmasstheyaredepressed.Thus,thecyclonehasacoldcorebelowandawarmcoreabove.3.AnairmassofrelaCvelylowpotenCalvorCcitygivesrisetoananCcyclone.ThedeviaConsinpressureandtemperaturehavetheoppositedirecConalsensetothoseincyclones.

Box1.12

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Character of the air mass in relation to PV

Today the low PV-air masses at 320 K are replaced by a high PV-air mass: see following slides

FromthesoluConofthePV-inverCbilityprinciplewemayalsoconcludethat:

acoldairmassonlyremainscoldaslongastherearemassesofhighpoten<alvor<cityaboveormassesofreducedpoten<alvor<citybelowit.Whenthiscondi<onisnolongerfulfilled,theairsinksdownandlosesthecharacterofacoldairmass.

Backgroundliterature:hep://www.staff.science.uu.nl/~delde102/HMR[1985].pdf

Box1.12

VerCcalsecConalongthelinea-ainthemapbelow,throughacut-offcycloneovernorth-westRussiaonNov.16,1959,12UTC.Isentropesareshownbydashed.lines.

FromPalmenandNewton,1969.

Note:thedownwardbulgingisentropesinthetroposphereandupwardbulgingisentropesinthestratosphere,associatedwithaposi<vePV-anomalyatthetropopause

Dashed:500hPatemperature:

Example: analysis of a cut-off low SeeweatherdiscussionMichielBaatsen23november2016

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Problem 1.26

850hPa,3March1995,00UTC

DeterminethegeostrophicwindatDeBiltfromobservaConsofthegeopotenCalheightatthreestaConsinthe“vicinity”bylinearinterpolaCon.

GeopotenCal,DeBilt,1March–5March1995

Blue:observedRed:analysed

Remarkablygoodanalysis!!

Pythonscript:seetheappendix

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BaroclinicRossbywave

BaroclinicRossbywave

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BaroclinicRossbywave

Cyclogenesis

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Tutorial25November

Blue:observedRed:geostrophic

Wind,DeBilt,1March–5March1995

passageofstorm

ageostrophicwind

Thegeostrophicwindspeedfollowstheobservedwind,exceptduringthepassageofthedepressiononMarch3,00UTC,andalsoonMarch5,00UTC.

Ingeneralgeostrophicwindspeedisgreaterthantheobservedwind!Why?

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AgeostrophicwindSecCon1.31:Devia<onsfromgeostrophy

€

d! v dt

= − f ˆ k × ! v −! ∇ Φ

€

f ˆ k × ! v g = −! ∇ Φ

geostrophicwind:

AgeostrophicwindSecCon1.31:Devia<onsfromgeostrophy

€

d! v dt

= − f ˆ k × ! v −! ∇ Φ

€

f ˆ k × ! v g = −! ∇ Φ

geostrophicwind:

€

! v = ! v g +! v a

Ageostrophicwind

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AgeostrophicwindSecCon1.31:Devia<onsfromgeostrophy

€

d! v dt

= − f ˆ k × ! v −! ∇ Φ

€

f ˆ k × ! v g = −! ∇ Φ

geostrophicwind:

€

! v = ! v g +! v a

Ageostrophicwind

€

d! v dt

= − f ˆ k × ! v a

€

! v a =1f

ˆ k × d! v dt⎛ ⎝ ⎜

⎞ ⎠ ⎟

AgeostrophicwindSecCon1.31:Devia<onsfromgeostrophy

€

! v = ! v g +! v a

€

d! v dt

= − f ˆ k × ! v a

€

! v a =1f

ˆ k × d! v dt⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

d! v dt

= − f ˆ k × ! v −! ∇ Φ

Ageostrophicwind

€

f ˆ k × ! v g = −! ∇ Φ

geostrophicwind:

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AgeostrophicwindSecCon1.31:Devia<onsfromgeostrophy

€

! v = ! v g +! v a

€

d! v dt

= − f ˆ k × ! v a

€

d! v dt

= − f ˆ k × ! v −! ∇ Φ

€

! v a =1f

ˆ k × ∂∂t

1f

ˆ k ×∇Φ⎛

⎝ ⎜

⎞

⎠ ⎟ +∂! v a∂t

+! v ⋅ ∇( )! v +ω∂

! v ∂p

⎛

⎝ ⎜

⎞

⎠ ⎟

Iner<al-advec<vewindIsallobaricwind

Ageostrophicwind

€

f ˆ k × ! v g = −! ∇ Φ

€

! v g =1f

ˆ k ×! ∇ Φ

geostrophicwind:

€

! v a =1f

ˆ k × d! v dt⎛ ⎝ ⎜

⎞ ⎠ ⎟

Isallobaricageostrophicwind

Isallobaricwind

Ageostrophicwind

Isallobaricwindcansome<mesbeaveryimportantcontribu<on:nextslide

Isallobaricwindblowsfromposi<vegeopoten<altendenciestonega<vegeopoten<altendencies

Iner<al-advec<vewind(duetoe.g.curvatureoftheflow)

However:theinerCal-advecCvewindusuallycouteractstheisallobaricwind

€

! v = ! v g +! v a

€

! v a =1f

ˆ k × ∂∂t

1f

ˆ k ×∇Φ⎛

⎝ ⎜

⎞

⎠ ⎟ +∂! v a∂t

+! v ⋅ ∇( )! v +ω∂

! v ∂p

⎛

⎝ ⎜

⎞

⎠ ⎟

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-6.3hPa/3hrs+3.2hPa/3hrs

isallobaricwind:

sea-levelisobars:

Jetstreaksandeddyac<vity

m

staConaryjetstreak

staConaryjetstreak

maximumineddy/wave-acCvityinlebexitofjetstreak

maximumineddy/wave-acCvity

300hPa SecCon1.31

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Jetstreak

(figure1.88)

yx

€

dudt

> 0

€

va > 0

€

va < 0

€

! v a =1f

ˆ k × d! v dt⎛ ⎝ ⎜

⎞ ⎠ ⎟

€

dudt

< 0

Next

Wednesday30November2016:

Problems1.29

Workonproject2

Friday2December2016:

DeviaConsfrombalance:“ageostrophiceffects”

Jetstreaks

Ekmanlayer

Rossbywaves

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Appendix: Python script problem 1.26 importnumpyasnpimportmatplotlib.pyplotaspltimportmathasmathdirec="/Users/Shared/DYME/"#setworkingdirectoryfile=open(direc+"Table1_5AD.txt","r")#createafileobject(secCon5.1Lin)lines=file.readlines()#readtheenCrefile(secCon5.2.1Lin)file.close()#closethefileobjectday=[]#createalisthour=[]z0=[]z1=[]z2=[]z3=[]uabs=[]

foriinrange(0,len(lines)):line=lines[i].split()#Breaksstringintoalist(theblankspacesinthestringareidenCfiedasdelimiters)(secCon5.2.3Lin)day.append(float(line[0]))#appendallelementsinthefirstcolumnofthefileobjecttothelistdayhour.append(float(line[1]))z0.append(float(line[2]))z1.append(float(line[3]))z2.append(float(line[4]))z3.append(float(line[5]))uabs.append(float(line[6]))

c=math.cos(52.1*math.pi/180.)s=math.sin(52.1*math.pi/180.)g=9.81omega=7.292*0.00001f=2*omega*s

Python script problem 1.26, continued y=[52.1,52.68,50.79,53.38]#laCtudeofDeBiltHembsyUccleEmdenx=[5.18,1.67,4.34,7.23]#longitudeofDeBiltHembsyUccleEmden

M=np.zeros((3,3),dtype='d')#matrix,M:rows,columns

M[:,0]=1.0foriinrange(3):M[i,1]=(x[i+1]-x[0])*111.*1000.*cM[i,2]=(y[i+1]-y[0])*111.*1000.

A=np.linalg.inv(M)#inverseofM

Cmeaxis=np.zeros((len(lines)),dtype='d')uabsg=np.zeros((len(lines)),dtype='d')phi=np.zeros((len(lines)),dtype='d')phi_obs=np.zeros((len(lines)),dtype='d')

forntinrange(len(lines)):Cmeaxis[nt]=day[nt]+hour[nt]/24.z_obs=[z0[nt],z1[nt],z2[nt],z3[nt]]phi_obs[nt]=g*z0[nt]phi[nt]=0.dphidx=0.dphidy=0.foriinrange(3):phi[nt]=phi[nt]+(A[0,i]*g*z_obs[i+1])dphidx=dphidx+(A[1,i]*g*z_obs[i+1])dphidy=dphidy+(A[2,i]*g*z_obs[i+1])

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Python script problem 1.26, continued ug=-(1./f)*dphidyvg=(1./f)*dphidxuabsg[nt]=math.sqrt((ug*ug)+(vg*vg))uabs[nt]=uabs[nt]*0.514#convertfromknotstom/s

print"z_ana=%6.1f,ug=%3.1f,vg=%3.1f,uabsg=%3.1f,uabs_obs=%3.1f"%(phi[nt]/g,ug,vg,uabsg[nt],uabs[nt])

plt.xlabel('date(March1995)')plt.ylabel('geopotenCalDeBilt(analysed:red;observed:blue)')plt.plot(Cmeaxis,phi,color='red')plt.plot(Cmeaxis,phi_obs,color='blue')plt.show()

plt.xlabel('date(March1995)')plt.ylabel('windspeed[m/s](geostrophic:red;observed:blue)')plt.plot(Cmeaxis,uabsg,color='red')plt.plot(Cmeaxis,uabs,color='blue')plt.show()

Input-file: Table1_5AD.txt 0100143513951475141352011213611316139713525302001348130813791343410212133913161358131628030012561140127913014903121259125112931214380400131913311340128116041213721365138213531005001338126813601343290512123412171264123631