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Dynamic Meteorology 2016 (Exercise session 8)

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

Topics

Evaluation of the weather forecast of 30 October 2016 by “4-cast”

Introduction to project 2 (couples; hypothesis due next 18 Nov.)

Weather discussion by Michiel Baatsen

Exercises from lecture 7, in particular problem 1.18

DynamicMeteorology(Project2)

(h9p://www.staff.science.uu.nl/~delde102/dynmeteorology.htm)

Introduction to project 2 (problem 1.24, p. 152) Principal component analysis (Box 1.11)

Forinspira*on:readsec*ons1.26-1.29

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IntroducEontotheproject

FormulateahypothesisabouttheconnecEon(correlaEon)betweenthree(ormore)differentatmospheric(oroceanic)variables(aso-calledateleconnec*on)

ExampleofateleconnecEon:(1)  RelaEonbetweenwintertemperatureinWesternEurope,wintertemperature

inEasternCanadaandsea-levelpressuredifferencebetweenIcelandandtheAzores(box1.11)

(2)  RelaEonbetweensealevelpressureinDarwinandsealevelpressureinTahiE(nextslide)

SendmeyourhypothesisbeforeoronFriday18November2016.

Collectdata(fromtheinternet)ofthesevariables.InvesEgatethisconnecEonwithastaEsEcalanalysistechniquecalledprincipalcomponentanalysis(box1.11).

Evaluateanddiscusstheresultsinanoralpresenta:onofabout15minutesonWednesday18January2016.

(Incouples)

TropicalteleconnecEon

J.Bjerknes,1969:AtmosphericTeleconnecEonsfromtheequatorialPacific.MonthlyWeatherReview,97,163-172.

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FigurefromWallaceandGutzler,1981:Mon.Wea.Rev.,109,784

“BeltsofAcEon”

“Teleconnec*on”betweenArcEcandlowermiddlelaEtudes

NorthernhemisphereteleconnecEonOne-pointcorrelaEonmapforthenorthernhemisphere

Cross-correlaEonmatrixsealevelpressure

FIGURE1.58.CorrelaEonbetweenzonal-meanandmonthlymeansealevelpressureanomaliesinthenorthernhemisphere(1958-2000).Thecontourintervalis0.3forposiEvevaluesand0.15fornegaEvevalues.NegaEvevalueslowerthan-0.3areshaded.Notethatthemapissymmetricaboutthediagonal,asexpected.Source:J.LiandJ.X.L.Wang,2003:Amodifiedzonalindexanditsphysicalsense.Geophys.Res.LeM.,30,1632.

DefiniEonoftheNAM-indexisbasedonthecross-correlaEonmatrixofmonthlymean(mm)zonalmean(zm)valuesofsealevelpressure(slp)inthenorthernhemisphere.HighestcorrelaEonisfoundformm,zmslp-valuesat65°Nand35°N(forDJF)

NorthernAnnularMode(NAM)

GraphicalrepresentaEonofcross-correlaEonmatrix

65°N

35°N

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Averageseasonalcycleofzonalmeanmonthlymeansea-levelpressure

FIGURE1.54.Annualmarchofthezonalmeanandmonthlymeansealevelpressureat36°Nandat66°N,averagedovertheyears1979-2012.BasedonERA-Interimreanalysis(h9p://data-portal.ecmwf.int/data/d/interim_moda/).

Zonalmean,monthlymeansealevelpressureanomaliesat66°Nand36°NinDecember–March(phase2)1979-2012(136months)

ERA-Interim1979-2012

PosiEveanomaliesrepresenttheposiEvephaseoftheNorthernAnnularMode(NAM),whilenegaEveanomaliesrepresentthenegaEvephaseofNAM

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NorthernAnnularModeIndex

FIGURE1.68.Plotofthemonthlymeannorthernhemispherezonalindex(theNAMindex)asfuncEonofEmefrom1873to2011.Source:h9p://ljp.lasg.ac.cn/dct/page/65544.

Definedasthenormalized(dividedbythestandarddeviaEon)differencebetween35°Nand65°Nofthezonalmeansealevelpressureanomalies.

LowpressureinArc9c

HighpressureinArc9c

LongtermvariaEonmayhavetodowithocean(AMO)(figure1.69)

ResearchquesEonInhowfarisincreaseofjanuarytemperatureduetoincreaseoftheNAM-index?

IncreaseoftemperatureinDeBiltmaybefullyexplainedbyenhancedgreenhouseeffect.

Box1.11

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Hypothesis

Hypothesis:LowwintertemperaturesinCanadaareassociatedwithhighAnnularModeIndex(lowsea-levelpressureinArc*c)andhighwintertemperaturesinCanadaareassociatedwithlowAnnularModeIndex(highsea-levelpressureinArc*c)

Physicalinterpreta*on:WintertemperatureinWesternEuropeisrelatedtoaveragestrengthofwesterlywind

Averagestrengthofwesterlywindisdeterminedbythemeridionalgradientofsea-levelpressureovertheNorthAtlanEcOcean,i.e.bytheNorthernAnnularModeIndex

IfpressureisrelaEvelowovertheArcEcandNorthAtlanEc,westerlywindisstrong,temperaturesarerelaEvehighiswesternEuropeandrelaEvelowinEasternCanada.

Example:

Data

Collectdata:

TimeseriesofmonthlymeantemperatureinwinterinWesternEurope,inEasternCanadaandcorrespondingEmeseriesofmonthlymeanNAM-index

Example:

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CorrelatethreeEmeseries:temperatureDeBilt,temperatureGooseandNAM-Index

w

Therefore,lowwintertemperaturesinGoose(Canada)areassociatedwithhighannularmodeindex(lowsea-levelpressureinArcEc),whilehighwintertemperaturesinCanadaareassociatedwithlowannularmodeindex(highsea-levelpressureinArcEc).AretemperaturesGooseandDeBiltreallyan*-correlated??

Goose(Canada)isatthesamelaEtudeasDeBilt

Jan.1950-2011Jan.1950-2011

CorrelatethreeEmeseriesExample:

TemperatureDeBiltisanEcorrelatedwithtemperatureGoose,butnotvery.

Goose(Canada)isatthesamelaEtudeasDeBilt

TheupwardtrendinNAM-indexshouldinduceanegaEvetrendintemperatureatGoose,whichisnotthecase

Jan.1950-2011Jan.1950-2011

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PrincipalComponentAnalysis(PCA)maybringsomeclarity

Thisanalysis(redline)representsaPCAontwovariables

€

y = 0.48x + 2.1FirstPrincipalComponent(PC):

Explains57%ofthetotalvariance

HowdowedothisifwehavemorethantwoEmeseries?

€

M =

1.000 0.5669 −0.50360.5669 1.000 −0.3217−0.5036 −0.3217 1.000

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

NAM DeBilt Goose

DeBilt

Goose

NAM

Cross-correlaEonmatrix:

PrincipalComponentAnalysis(PCA)maybringsomeclarity

Thisanalysis(redline)representsaPCAontwovariables

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y = 0.48x + 2.1FirstPrincipalComponent(PC):

Explains57%ofthetotalvariance

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Eigenvaluesofcross-correlaEonmatrixExample:

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! x 1 ≡ −0.630, −0.565, 0.533( )

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! x 2 ≡ 0.077, 0.638, 0.766( )

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! x 3 ≡ 0.773, −0.524, 0.358( )

λ1=1.935; λ2=0.682; λ3=0.383

First eigenvector explains 64.5% of total variance in data

NAM T,DeBilt T,Goose

64.5%

22.7% CO2??

Eigenvectors:

12.8%

NegaEveNAMiscorrelatedwithnegaEvetemperatureanomalyatDeBiltandposiEvetemperatureatGooose

Projectdataonfirsteigenvector(PC)Example:firstprincipalcomponent

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! x 1 ≡ −0.630, −0.565, 0.533( ) 64.5%

NAM DeBilt Goose

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Reanalysiswebsites:h9p://apps.ecmwf.int/datasets/h9p://www.esrl.noaa.gov/psd/data/gridded/data.ncep.reanalysis2.htmlh9p://www.esrl.noaa.gov/psd/data/Emeseries/h9p://disc.sci.gsfc.nasa.gov/daac-bin/FTPSubset.pl

Data-sources

import numpy as Nimport netCDF4 as Simport matplotlib.pyplot as P

year_len = 36.0mon_len = 12.0

#open filea = S.Dataset("T2m-mm-52N05E.nc", mode='r') #ERA-Interim data: temperature at 2 m at 52N and 5E (one grid point)

#create arrayslat = a.variables["latitude"][:] lon = a.variables["longitude"][:] time = a.variables["time"][:]time = year_len * (time - N.min(time)) / ( N.max(time) - N.min(time))+1979

# T2mT2m = a.variables["t2m"][:,:,:]

y = T2m[:,0,0] # make one-dimensional array

P.plot(time, y)P.scatter(time, y, s=20, color='red', marker=u'o') P.axis([1979,2014,270.0,300.0]) # define axes P.xlabel('year') # label along x-axesP.ylabel('Temperature at 2 m [K]') # label along x-axesP.title('Monthly mean temperature at 52N, 0E') # Title at top of plotP.grid(True)P.show() # show plot on screen

Pythonscripttoreadandplotanetcdf-filewhichhasbeendownloadedfromh9p://apps.ecmwf.int/

Theresult:

NextFriday,4November2015,11:00-12:45BBG201

LectureontheuseofpressureorpotenEaltemperatureasaverEcalcoordinate(isobaricorisentropiccoordinates)

PotenEalvorEcityinisentropiccoordinates

IsentropicpotenEalvorEcitymixingby“planetarywaves”

Week45:examonFriday11November,9-12,roomBBG209.

TheexamisaboutsecEons1.1-1.22(exceptsecEon1.13),andalsoboxes1.1and1.2ofthelecturenotes(thepartthathasbeencoveredinthefirst7lectures).YoudonothavetoknowequaEonsbyheart,butitisofcoursegoodtobefamiliarwithequaEons,sothatyoucanrecognisethetermsandinterprettheequaEon.

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PROBLEM 1.18 An approximate model of the horizontal distribution of velocity in a tropical cyclone (box 1.6) is the so-called "Rankine vortex". This is an axisymmetric circular vortex with an azimuthal velocity, vθ, which is a function of the radius, r (the distance from the centre of the vortex), as follows.

Here v0 is the maximum wind velocity and R is the radius of maximum wind velocity.

(a)  Calculate and plot the relative vorticity as a function of r for a Rankine vortex with v0=40 m s-1 and R=40 km.

(b)  Estimate the inertial period “inside” the radius of maximum wind.(c)  Is the inertial stability in the core of hurricane Alicia large or small?

€

vθ =v0rR

for r ≤ R and vθ =v0Rr

for r > R