Quasi-balance and quasi-geostrophic approximation (section 1.31) Balanced circular vortex. Example: tropical cyclone (section 1.32) Potential vorticity (PV) inversion equation (section 1.33) Structure of balanced circular vortex from solution of PV-inversion equation

DynamicMeteorology:lecture9

Sections 1.27,1.28 1.31, 1.32, 1.33

NOTUTORIALONWEDNESDAY7/11/2018

Problems1.29and1.30,1.31and1.32

14/11/2018: tutorial 8

2/11/2018 (Friday)

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

Quasi-balance and quasi-geostrophic approximation

Anexample:3March1995,12UTC,500hPa

Following3slidesshow:(1)Observedwindinterpolatedtoaregular“lat-lon”grid(2)ObservedgeopotenWalheightinterpolatedtoaregular“lat-lon”grid(3)GeostrophicwindcalculatedfrominterpolatedobservaWonsofgeopotenWalheight

(1)Observedwindinterpolatedtoaregular“lat-lon”grid

Seefigure1.86

3March1995,12UTC,500hPa

Measuringpoint

(2)Observedwindandobservedheight,500hPa,interpolatedtoaregular“lat-lon”grid

Seefigure1.86

3March1995,12UTC,500hPa

Labelsinunitsofm

Measuringpoint

(3)GeostrophicwindcalculatedfrominterpolatedobservaWonsofheight,500hPa

3March1995,12UTC,500hPa

Seefigure1.86

Labelsinunitsofm

Measuringpoint

(1)Thewindvelocityisequaltothesumofthebalanced(“geostrophic”)wind(subscriptg)andthedeviaWonfromthisbalancedwind(subscripta,standingfor“ageostrophic):

€

! v = ! v g +! v a

€

! v a <<! v g

€

f = f0 +β y− y0( )(2)TheCoriolisparameterisalinearfuncWonoflaWtude:

wheref0isthevalueoffatachosenreferencelaWtude,y=y0(usually45°NorS),andwheretheso-called“β-parameter”isdefinedby

€

β ≡dfdy

at y = y0

€

f0 >> β y − y0( )

Quasi-geostrophic approximation

(3)Thequasi-geostrophicapproximaWonisusuallyappliedtotheequaWonsinpressurecoordinates.InthebalancedstatethematerialderivaWveandthecurvaturetermsinequaWonsofmoWon(1.191a,b)areneglected,sothatthebalanced(“geostrophic”)windisapproximatedby

€

ug ≈ −1f0∂Φ∂y

€

vg ≈1f0∂Φ∂x

section 1.31

Quasi-geostrophicvorFcity,ζg,canbeexpressedintermsofthegeopotenFal,using(1.246),as

€

ζ g ≡∂vg∂x

−∂ug∂y

=1f0

∂2Φ

∂x2+∂2Φ

∂y2⎛

⎝ ⎜

⎞

⎠ ⎟ ≡

1f0∇h2Φ

quasi-geostrophic vorticity

(1.249)

€

∇h2Φ = f0ζg

ThegeopotenFal,Φ,isdeterminedby“inverFng”thePoisson(ellipFc)equaFon:

€

ug ≈ −1f0∂Φ∂y

€

vg ≈1f0∂Φ∂x

Geostrophicwindaccordingtothequasi-geostrophicapproximaWon:

(1.246)

f assumedconstant!

Example: hurricane “Katrina”

Nextslide:cross-secWonalongthislaWtude

Figure1.95:HurricaneKatrina(2005)at925hPa

Blueshadingindicatesareawhereζ>2×10-4s-1.

€

∇h2Φ = f0ζg

BalanceequaWon:

QualitaWvelyOK,butnotalwaysquanWtaWvely!(problem1.31)

Cross-section through cyclone “Katrina”

€

∇h2Φ = f0ζg

Figure1.96

(1.253)

aminimumingeopotenFalcorrespondstoamaximuminrelaFvevorFcity

Φ/g [m]

West-eastcrosssecWonthroughcentreof“Katrina”at925hPa

Inpolarcoordinatesandassumingaxisymmetry(1.253)is:

€

1r∂∂r

r ∂Φ∂r

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = f0ζg

r

ζ [s-1]Givenζ,Φandthebalancedazimuthalwindvelocity(seenextslide)isdeterminedby“inverWng”eq.1.253.

Figure1.49

Gradient wind balance in a hurricaneHurricane“Alicia”

TheWmemeanaxisymmetricswirling(azimuthalortangenFal)wind(solidline)andthegradient(balanced)wind(crosses)(explainedinsecFon1.32)atthe850hPapressurelevel,measuredintropicalcyclone(hurricane)Alicia(28°N,94°W)intheWmeinterval11:59to18:18UTC,August17,1983.

€

u2

r+ fu− ∂Φ

∂r= 0

u

r

Inpressurecoordinates:

“balancedswirlingwind”

€

1r∂∂r

r ∂Φ∂r

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = f0ζg

Quasi-geostrophic:

Gradientwindbalance:

AboveequaWonsarenotcompletelyconsistent.Why?

BejerapproximaWontobalanceinahurricane:

The structure of quasi-balanced vortices

(1)polarvortex:thecycloniccirculaWoncentredoverthepoleinwinter(figures1.40&82);(2)baroclinicmiddlelaFtudecyclone:typicalfrontalcyclonebelowthejet(figures1.116,118&119);(3)cold-corecut-offcycloneintheuppertroposphere(figures1.102&103);(4)tropicalcyclone:awarmcoreintensecycloneinthetropics(Box1.11&figures1.95&97);(5)warm-coresubtropicalanF-cyclone:suchastheAzoreshigh(figure1.37);(6)warm-coreblockinganF-cyclone:highlaWtude“cut-off”high(figure1.114);(7)cold-coreconFnentalanF-cyclone:Asianwinterhigh(figure1.37).(8)polarlow:warmcorecycloneoccurringoverthehighlaWtudeoceans,suchastheNorwegiansea(9)“medicane”:awarmcorecycloneoccurringoverthewarmMediterraneanseainAutumn(10)lee-cyclone:occurs,forexample,intheGulfofGenoainanortherlyair-flowovertheAlps

Whattypeisthis?

project0*:writeanessayaboutacasestudyonthelifecycleandstructureofanatmosphericvortex

6Oct.2018,14:30UTC7Oct.2018,11:15-13:00UTC

*Page13(lecturenotes)

Idealised (axi-symmetric) vortex: in gradient wind balance

u>0

r

u<0

Radialwind:

Swirlingwind,u:

Balanceinanaxisymmetricvortexinisentropiccoordinatescanbeexpressedas

SecFon1.33

€

u2

r+ fu− ∂Ψ

∂r= 0

FpFcor+Fcen Fp+Fcen Fcor

Cyclone AnFcyclone

*

*note:wehavenotansweredthequesWonwhyairparcelsshouldflowincircularpaths

Axi-symmetric vortex in thermal wind balance

HydrostaFcbalance:

Thermalwindbalance:

Gradientwindbalance:

eliminateΨ

SecFon1.33

}

Thermalwindbalance:

Axi-symmetric vortex in thermal wind balance SecFon1.33

Thermal wind balance leads to potential vorticity inversion

f=constant

Previousslide:

(thermalwindbalance)

(PV-inversionequaFon)

SecFon1.33

Anotherwayofexpressingthermalwindbalance

Potential vorticity inversion

(PV-inversionequaFon)

PV-inversionequaWonisanellipWcparWaldifferenWalequaWonif

InthatcasethisequaWonhasauniquesoluWon,ifboundarycondiWonsarespecified.

YoucanfinduifyouknowZ(andu attheboundariesofthedomainofinterest).

SecFon1.33

Boundary conditions

Whatdoesthischoicemean?

ThestandardformulaWonoftheboundarycondiFonforthesoluFonofanellipFcsecondorderparFaldifferenFalequaFonistospecifyuontheboundary(Dirichletboundarycondi1on)ortospecifythenormalderivaWveofuontheboundary(Neumannboundarycondi1on).Intheproblemathand,weimposeaDirichletboundarycondionatthepole,attheupperboundaryandat10°N.Atthelowerboundary,whichisnotasmoothcurve,weimposeanumericalapproximaWonoftheNeumannboundarycondiWonbyprescribing∂u/∂θ.Becauseofthenon-linearityofthePV-inversioneq.andbecauseofthecomplexmixedboundarycondi1ons,thesoluWonofthisequaWonisfarfromastandardmathemaWcalproblem.

SecFon1.33

(PV-inversionequaFon)

Elliptic equations appear when some kind of balanced state is assumed

EXAMPLE:Geostrophicbalanceinpressurecoordinates(seeslide5):

f=f0=constant

ThisisPoisson’sequaWon,whichisalsoanellipWcequaWon.

DescribestherelaWonbetweenstreamfuncWonandrelaWvevorWcityinabalancedatmosphere(similartoeq.1.253).

SecFon1.33

€

ζg ≡∂vg∂x

−∂ug∂y

=∂∂x

1f0∂Ψ∂x

⎧ ⎨ ⎩

⎫ ⎬ ⎭

+∂∂y

1f0∂Ψ∂y

⎧ ⎨ ⎩

⎫ ⎬ ⎭

=1f0∇2Ψ

€

∇2Ψ = f0ζg

potential vorticity inversion PV-inversionequaWon:

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.

PosiFvePV-anomaly

NumericalsoluFon(methodisdescribedinchapter7)SecFon1.33

f=constant!

€

Zref θ( ) =f

σ ref θ( )

potential vorticity inversion PV-inversionequaWon:

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.

NegaFvePV-anomaly

NumericalsoluFon(methodisdescribedinchapter7)

f=constant!

€

Zref θ( ) =f

σ ref θ( )

SecFon1.33

Solution due to Kleinschmidt (1957)*

Acyclone“produced”byabodyof6WmesthenormalpotenWalvorWcity.Theundisturbedreferencestateconsistsoftwolayerswithaconstanttemperaturelapserate:atropospherewithdT/dz=5.8°Ckm-1andanisothermalstratosphere.Thelew-handdiagramshowsthetemperatureontheaxis(Ta)andintheundisturbedatmosphere(Tu)asafuncWonheight.Cross-secWon:thinlinesareisentropeslabeledinK;heavylinesintheleehalfindicatetherelaFvedepression(pu-p)/pu(perthousand)(puisthepressureintheundisturbedatmosphere).Heavylinesontherightareisotachs,labeledinms-1.

SecFon1.33

*Referenceinlecturenotes,page197,figure1.101.

Solution due to Keinschmidt (1957)

GeneralconclusionfrompotenWalvorWcityinversion:

1a.WithinanisolatedairmasswithabnormalpotenFalvorFcity,inthermalwindbalance,theabsolutevorFcitydeviatesfromthenormalinthesamesenseasthepotenFalvorFcity.

1b.WithinanisolatedairmasswithabnormalpotenFalvorFcity,inthermalwindbalance,theisentropicdensitydeviatesfromthenormalintheoppositesenseasthepotenFalvorFcity.

SecFon1.33

PV+σ-ζ+

Solution due to Keinschmidt (1957)

SecondgeneralconclusionfrompotenWalvorWcityinversion(relatedtoconclusion1abonpreviousslide):

2.AnairmassofrelaWvelyhigh(low)potenWalvorWcitygivesrisetoacyclone(ananWcyclone).

SecFon1.33

PV+

Solution due to Keinschmidt (1957)

ThirdgeneralconclusionfrompotenWalvorWcityinversion:

3.IsentropicsurfacesaredepressedaboveanairmassofrelaFvelyhighpotenFalvorFcityinthermalwindbalance.IsentropicsurfacesareraisedbelowanairmassofrelaFvelyhighpotenFalvorFcityinthermalwindbalance.Thus,anupperlevelcyclonehasacoldcorebelowandawarmcoreabove,whileanupperlevelanWcyclonehasawarmcorebelowandacoldcoreabovethepotenWalvorWcityanomaly

SecFon1.33

warm

cold

σ+

σ+

CycloniccirculaWonaroundaposiWvePV-anomaly

AnWcycloniccirculaWonaroundanegaWvePV-anomaly

cold

warm

warm

cold

Character of the air mass in relation to PV

FourthgeneralconclusionfrompotenWalvorWcityinversion:

(4)Acoldairmassinthermalwindbalance,onlyremainscoldaslongastherearemassesofhighpotenFalvorFcityaboveormassesofreducedpotenFalvorFcitybelowit.WhenthiscondiFonisnolongerfulfilled,theairsinksdownandlosesthecharacterofacoldairmass.

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

SecFon1.33

Gradient wind balance of the zonal mean circumpolar circulation (section 1.34) Acceleration of the zonal mean zonal wind by eddies (section 1.35) Quasi-geostrophic vorticity equation (section 1.31) Planetary Rossby waves (section 1.37)

DynamicMeteorology:lecture10

Sections 1.34,1.35,1.36,1.37 (recap of 1.31)

Problems1.33and1.34

21/11/2018: tutorial 9

Next : 16/11/2018 (Friday) Lecture by Michiel Baatsen