Detection of Coherent Airstreams Using Cluster Analysis

14 Pages·2015·3.29 MB·English

Detection of coherent airstreams using
cluster analysis: application to an
extratropical cyclone
Article
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Hart, N. C. G., Gray, S. L. and Clark, P. A. (2015) Detection of
coherent airstreams using cluster analysis: application to an
extratropical cyclone. Monthly Weather Review, 143 (9). pp.
3518-3531. ISSN 1520-0493 doi:
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3518 MONTHLY WEATHER REVIEW VOLUME143
Detection of Coherent Airstreams Using Cluster Analysis: Application to an
Extratropical Cyclone
NEILC.G.HART,SUZANNEL.GRAY,ANDPETERA.CLARK
DepartmentofMeteorology,UniversityofReading,Reading,UnitedKingdom
(Manuscriptreceived17November2014,infinalform8April2015)
ABSTRACT
Flow in geophysical fluids is commonly summarized by coherent streams (e.g., conveyor belt flows in
extratropicalcyclonesorjetstreaksintheuppertroposphere).Typically,parceltrajectoriesarecalculated
from the flow field and subjective thresholds are used to distinguish coherent streams of interest. This
methodology contribution develops a more objective approach to distinguish coherent airstreams within
extratropicalcyclones.Agglomerativeclusteringisappliedtotrajectoriesalongwithamethodtoidentifythe
optimalnumberofclusterclasses.Themethodologyisappliedtotrajectoriesassociatedwiththelow-leveljets
ofawell-studiedextratropicalcyclone.Forcomputationalefficiency,aconstraintthattrajectoriesmustpass
through these jet regions is applied prior to clustering; the partitioning into different airstreams is then
performedbytheagglomerativeclustering.Itisdemonstratedthatthemethodologycanidentifythesalient
flowstructuresofcyclones:thewarmandcoldconveyorbelts.Atestfocusingontheairstreamsterminatingat
thetipofthebent-backfrontfurtherdemonstratesthesuccessofthemethodinthatitcandistinguishfinescale
flowstructuresuchasdescendingsting-jetairstreams.
1. Introduction archetypes is useful. Cheng and Wallace (1993) identi-
fiedlarge-scaleatmosphericflowregimesbyapplyinga
Thestructureofawiderangeofgeophysicalflowshas
hierarchical clustering approach to 500-hPa geo-
often been analyzed in terms of distinct, coherent air-
potential height fields. Hierarchical clustering has also
streams, such as jet streams, jet streaks, and conveyor
beenusedtocategorize synoptic-scalerainfallpatterns
belts.Forexample,aconveyorbeltviewofflowwithin
from a high-density rain gauge network (e.g., Tennant
extratropicalcyclonesiswidelyaccepted(Harrold1973;
andHewitson2002;Crétatetal.2012).Fuzzyclustering
Carlson1980;BrowningandRoberts1994;Wernliand
approachessuchastheK-meansalgorithmhavebecome
Davies 1997; Schultz 2001). Though clearly defined
the favored methods for determining weather regimes
features in satellite imagery or synoptic-scale analyses,
because of the variety of advanced statistical tests that
the precise definition of these airstreams often relies
canbeusedtotesttherobustnessoftheregimesthatare
onarelativelyarbitrarychoiceofthreshold.Thegoalof
determined (e.g., Michelangeli et al. 1995; Fauchereau
thiscontributionistodemonstratethatclusteranalysis
etal.2009).
of flow trajectories is a successful method to automate
Application of cluster analysis to two-dimensional
the identification of coherent airstreams in a more
and later three-dimensional airflow trajectories was
objectiveway.
first carried out in research focusing on understanding
Cluster analysis has gained wide use in geophysical
variability in atmospheric composition at observations
sciences, particularly in applications where identifying
sites(e.g.,MoodyandGalloway1988;HarrisandKahl
1990). Both hierarchical and fuzzy clustering ap-
proacheshavebeenusedtocharacterizethetrajectories
computed backward from atmospheric composition
DenotesOpenAccesscontent.
observatories (e.g., Dorling et al. 1992; Moody et al.
1995; Cape et al. 2000). Stohl (1998) summarized the
Correspondingauthoraddress:NeilC.G.Hart,Departmentof
strengths and shortcomings of these trajectory compu-
Meteorology,UniversityofReading,EarleyGate,P.O.Box243,
tations and classification techniques. Despite this wide
ReadingRG66BB,UnitedKingdom.
E-mail:[email protected] useofclusteranalysis,weareunawareofanapplication
DOI:10.1175/MWR-D-14-00382.1
(cid:1)2015AmericanMeteorologicalSociety
SEPTEMBER2015 HART ET AL. 3519
to the flow in extratropical cyclones. There is an im- statistics across the airstreams has the caveat that the
portant difference between the application we present thresholdingmayadmitmoreorfewertrajectoriesinthe
and the literature noted above: most applications have WCB of each cyclone, resulting in statistical artifacts.
little a priori knowledge of the classifications cluster IfastudyfocusedonfinedifferencesbetweenWCBflow
analysis may produce, whereas extratropical cyclones in, for example an ensemble simulation of one storm,
haveawealthofliteraturedescribingairflowfeatures. thiscaveatcouldbecomeaseriousissue.Furthermore,if
The warm conveyor belt (WCB) is a warm moist theairstreamofinterestexhibitswidevariabilityintotal
(high-valued equivalent potential temperature) rain- ascent or descent, such as asting jet, thresholdcriteria
producing ascending airstream advancing poleward maywellpreventdetectionofvalidcoherentensembles
ahead of the cold front (Harrold 1973). The cold con- oftrajectories.Thiscontributionproposesclusteranal-
veyorbelt(CCB)isacoollow-levelairstreamthatforms ysisasanappropriatetoolinsuchasituation.Clustering
onthe cool side of thewarm front flowing rearwardin trajectories that pass near the frontal structures of
relationtocyclonemotion(Carlson1980;Schultz2001). extratropical cyclones should naturally result in co-
In extratropical cyclones where the warm front bends herentensemblesoftrajectories,basedonsimilaritiesin
cyclonically around behind the low pressure center of theirdynamicalhistories.Thesecouldthenbeclassified
the system, the CCB flow can wrap around to produce basedontheaprioriknowledgeofconveyorbeltflow;
very strong earth-relative winds immediately south of forexample,acoherentensembleoftrajectorieswould
thecyclonecenter.ThisiscommoninShapiro–Keyser- be identified as a WCB if it was a near-saturated as-
type cyclones (Shapiro and Keyser 1990), producing a cending airstream ahead of the cold front. The WCB
‘‘poisonous tail’’ of damaging winds (Grønås 1995). In could thus be selected without the need to choose a
such cyclones, finer-scale flow structures have some- thresholdcriterionotherthanatestofascent.
timesbeenfoundassociatedwithdamagingwindsahead The clustering method isdemonstratedhere inappli-
oftheCCB(Browning2004).Termed‘‘stingjets,’’since cationtothewell-observedandwell-studiedextratropical
they occur near thetip of thepoisonous tail, these air- cyclone, Cyclone Friedhelm (2011). Figure 1 provides
streamsdeveloplessfrequentlythantheCCBandWCB schematicrepresentationofwheretheairstreamsformed
(Martínez-Alvarado et al. 2012) and can have a more during the development of Cyclone Friedhelm, which
transient nature when they do develop, persisting for developed explosively and produced very strong and
periods of only several hours (Clark et al. 2005; Baker damaging surface winds over Scotland on 8 December
2009;Martínez-Alvaradoetal.2010). 2011 (Baker et al. 2013; Martínez-Alvarado et al. 2014,
Conveyor belt and sting-jet airstreams are often hereafterMA14).Duringtheearlystagesofdevelopment
identified subjectively or with simple thresholding theWCB wastheprimarycoherent ensembleoftrajec-
techniquesappliedtoLagrangiantrajectoriesdescribing tories associated with this cyclone (Fig. 1a). The CCB
these flows (Wernli and Davies 1997; Schultz 2001; becameassociatedwithstrongEarth-relativewindsasthe
Clarketal.2005).Thisapproachhasprovedparticularly warm front was cyclonically bent back around the low
effectiveforautomatingtheidentificationoftheWCBs pressurecenter (Fig.1b).Bythisstage thelow-level jet
in climatology studies of extratropical cyclones (Stohl associatedwiththeWCBwasstartingtoweaken.Asting
2001; Eckhardt et al. 2004; Madonna et al. 2014; Pfahl jet descended on the southern flank of the CCB as the
etal.2014).Trajectoriesthatstartnearthesurface,and cloudheadofthecyclonecontinuedtowraparoundwith
exceedatotalascentthreshold(e.g.,Dp.600hPa,where thebent-backwarmfront(Fig.1c).Whiletheschematic
pispressure)canberetainedasthecoherentensembles showstheevolutionoftheseflowsforCycloneFriedhelm,
oftrajectoriesdescribingtheWCB,asdemonstratedby this evolution generally occurs for developing cyclones
Wernli and Davies (1997). Madonna et al. (2014) note with diagnosed sting jets [Clark et al. (2005), e.g., the
that a 600-hPa ascent criterion (within a 2-day time GreatOctoberstormof1987].
period)is‘‘fairlystrong’’andfoundsomechangesinthe Ageneraldescriptionofthemethod,independentof
spatialdistributionofWCBstartingpointsandaverage applicationtoextratropicalcyclones,isgiveninsection
evolutionofparametervalues(suchasspecifichumidity 2. Section 3 describes the model simulation of Fried-
and potential vorticity) along the trajectories in sensi- helm, the cyclone used to demonstrate and test the
tivitytestsinwhichtheascentpressurechangeortime methodology. The results of applying this method to
criterionwererelaxed. thedominantlow-levelconveyorbeltairstreamsandthe
Applyingthresholdstoidentifyairstreamsworkswell mesoscale jet structure near the bent-back front are
where the airstream is largely similar across cyclone described in sections 3aand 3b, respectively.Section 4
populationsandhasaneasilyidentifiablecharacteristic provides a summary of these results and concludes
suchasstrongascentordescent.Nevertheless,calculating thisstudy.
3520 MONTHLY WEATHER REVIEW VOLUME143
FIG.1.Schematicofthedevelopmentofairstreams
andfrontlocationsinextratropicalcycloneFriedhelm
during8Dec2011overlaidontoinfraredimageryat
(a) 0300 UTC (MODIS), (b) 1000 UTC (AVHRR),
and(c)1300UTC(AVHRR).Thelightblueregion
denotes approximate region of strongest low-level
(e.g., 850hPa) winds. (Courtesy: Dundee Satellite
ReceivingStation).
2. Methodology:Agglomerativeclustering observationsintonewclusterclasses.Thealgorithmused
here is Ward’s variance minimization (Ward 1963); this
Agglomerative (also termed hierarchical) clustering hasastraight-forwardimplementation,whichlendsitself
depends on assessing the similarity between many in- toautomation,akeygoalofthisstudy.Itsprimarydraw-
dividual instances that are commonly referred to as backisthetendencytoproduceclusterclassescontaininga
observationsinclusteringalgorithms.Inthisapplication similarnumbersofobservations.Thisproducesthecaveat
each trajectory is one observation. These observations thatcoherentstreamscontainingmanytrajectoriesmaybe
arestoredinamatrixWcontainingJobservationseach described by more than one cluster class. The SciPy Hi-
describedbyNdimensions.Acommonmeasureofsim- erarchical Clustering module for Python (Jones et al.
ilarityisEuclideandistanced,computedasthel2 norm. 2001), used here, implements Ward’s method with the
BetweentwoobservationvectorsinWthiswouldbe widely used approach of updating a matrix storing the
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Euclidean distances between each cluster centroid
d 5 (cid:1)N [W(n)2W (n)]2, (1) (Wishart1969). Thealgorithm ensuresthat,at eachiter-
jk j k ation,newclassesarecreatedsuchthatvariancebetween
n51
members within a cluster class is minimized across all
wherejandkindexthetwoobservationsundercompari- possiblecombinationsofmembersofaclass,atthatstep.
son and n indexes the dimension of each observation. Iterationcontinuesuntilallobservationsareagglomerated
Clustering starts by agglomerating the most similar into a single class. Cheng and Wallace (1993) provide a
SEPTEMBER2015 HART ET AL. 3521
FIG.2.(a)Dendrogramindicatingthesuccessiveagglomerationofclustersfromobservations(xaxis)withanincreasingdistance
betweenclusters(yaxis),and(b)dasafunctionofagglomerationstep(blue)withthepeakvalueofcurvature(d2d/dx2,red)indicated
(dashed).Seethetextforadefinitionofd.Themeandistanceattheagglomerationstepwhenthepeakcurvatureoccursistheclassification
cutoffdistanceasshownbyahorizontaldashedlinein(a).Theresultingsixclustersarelabeledin(a).Panels(a)and(b)correspondtothe
clusteringoftrajectoriesinitializedat0600UTC(No.ofobservations59738)showninFig.4.(c),(d)Asin(a),(b),butcorrespondto
superclustering(No.ofobservations560)showninFig.6.
detailed account of the algorithm with additional de- N528, if the time period was over seven hourly
scriptions available in Crétat et al. (2012) and Ramos positions.
(2001). The succession of agglomerations is represented Scalingofthesevariablesisnecessarybecauseofthe
graphicallybydendrograms,asshowninFigs.2aand2c. different units of the horizontal, vertical, and airmass
Any distinguishingvariable canbe used in an obser- variables.Wechoosetoscalethevaluesofeachvariable
vationvector.Inthecontextofclassifyingatmospheric by their respective standard deviations at time t5t ,
0
flows, building the observation vectors from positional giving^x 5x/s[x (t )]andsimilarlyfory,z,andg.The
j j all 0
information [x(t), y(t), z(t), where t is time] is appro- matrixx representsamatrixofxpositionalvectorsfor
all
priate(e.g.,Dorlingetal.1992).Inclusionofanairmass alltrajectoriesandsrepresentsthestandarddeviation.
tracer variable [g(t)] can add further distinguishing in- Thus, ^x is the jth x-coordinate vector scaled with the
j
formation to the flows under consideration. Having standarddeviationofthexpositionofalltrajectoriesat
computed trajectories for a given flowfield (from start t5t .Scalinginsuchamannerforallvariablesineach
0
points at time t ) a time period of interest can be observation produces the data matrix that is passed to
0
chosen—t5[t 23, t 13h]—andtheobservationvector theclusteringalgorithm:
0 0
foroneofthesetrajectoriescanbespecified:
W^ 5[^x,^y,^y,g^ ]; j51:J. (3)
j j j j j
W 5[x,y,z,g], (2)
j j j j j
Twosubtletiestothesechoicesbearmention.First,the
whereboldvariablesdenoterowsofvaluesforthistime mean is not removed (if so this would be a normaliza-
period. This observation vector would have dimension tion)asthiswouldremovetheabilitytodistinguishthe
3522 MONTHLY WEATHER REVIEW VOLUME143
geographic locality of the trajectories, leaving only the time. This results in a population of system-relative
shape of each trajectory as the distinction. Second, class-median trajectories that summarize the weather
scalingbythestandarddeviationcalculatedonlyatthe system development. This summary class-median-
trajectorystarttimeensuresthattherelativetimeevo- trajectory population can then be classified with same
lution of each variable is unmodified (i.e., a trajectory clusteringapproachdescribedabovetoobtaina‘‘super’’
withsubstantialthree-dimensionalcurvatureretainsits classificationofairstreamsthatformasaweathersystem
characterinrelationtomorelineartrajectorypaths). evolves.InaShapiro–Keyser-typeextratropicalcyclone,
Thenatureofthetaskrequiresanapproachtoauto- thissuperclassificationshouldshowtheWCBflowduring
maticallydecideonthenumberofclusterstoretain.This early cyclone development and the CCB in later stages
decisionwouldideallyresultinaclassificationwithco- (Shapiro and Keyser 1990). This is demonstrated in
herent, dissimilar airstreams assigned to separate clas- section3a.
ses,andsimilarairstreamsgroupedinoneclass.Thiscan
beachievedbyexploitingafeatureoftheclusteringal-
3. Coherentensemblesoftrajectoriesinatestcase
gorithm itself. As agglomerative clustering proceeds,
cyclone
d between successive clusters joined at each iteration
increases gradually. This distance d can be averaged In this section we describe Cyclone Friedhelm and
overalltheclustersjoinedatoneiterationsteptogived. demonstrate that the method of clustering trajectories
Figure 2b shows this increase in d as a function of ag- can identify both synoptic-scale and mesoscale struc-
glomerationstep.Whennotablydistinctclustersstartto turesinitsflowfield.CycloneFriedhelmwasobserved
be cojoined d rapidly increases. The last set of cluster in situ using a research aircraft during intensive ob-
classes present before this sudden jump represents the servingperiod8oftheDiabaticInfluencesonMesoscale
classification of trajectories into classes most distinct Structures in Extratropical Storms (DIAMET) field
fromoneanother,asshownbythecurvatureinFig.2b. campaign(Vaughanetal.2015).Itwasthesubjectofa
Theseclassesaretheclassificationretainedasthesalient detailedcase studyinto theairstreamsthat constituted
airstreams in this methodology. Dorling et al. (1992) thelow-leveljetinthebent-backfrontalregion(MA14).
used a similar technique in application of a fuzzy clus- MA14 analyzed a numerical simulation that was com-
teringmethodfor2–30clusters.Thenumberofcluster paredtoflightdata.Theydefinedthelow-leveljetinthis
classeswaschosenasthatjustbeforeasuddenincrease region by the 45ms21 isotach and found three constit-
in intracluster variance, as expected when classes con- uent airstreams with distinct equivalent potential tem-
tainverydifferentclassmembers.This wasdecidedby peratures and airflow histories: a primary CCB flow, a
visual inspection (Dorling et al. 1992). In our method, secondaryCCBflow,andasting-jetdescent.Compari-
the decision of cluster numbers is automated, which son with MA14 provides a stringent test for this clus-
leads to the method admitting a caveat as follows. If a teringmethodology.
maximum in curvature occurs two agglomerations or The cyclone was simulated using the operational
morebeforeonlyoneclassisleft,themethodworksas numerical weather prediction model used by the Met
described; if not, the last value of the curvature is the Office,theUnifiedModel(MetUM).Version8.2ofthe
highestandthenumberofclasseschosenisforcedtobe MetUMwasusedwiththe(untilrecentlyoperational)
thenumberofclassespresentinthethird-lastiteration. North Atlantic and European domain configuration
ThisisborneoutinacomparisonofFigs.2band2d.At thatextendsfromapproximately308to708Ninlatitude
worst, this caveat results in more cluster classes being and from 608W to 408E in longitude (figures in this
selected than would have been selected by visual in- papershow asubregionofthisdomain).Thisconfigu-
spection. Therefore, no attempt is made to draw con- rationhas0.118(;12km)gridspacinginthehorizontal
clusionsfromthenumberofclasses. in both latitude and longitude on a rotated grid. The
Eachoftheseautomaticallychosenclassescontainsa model lid is at ;80km with the 70 stretched vertical
populationoftrajectoriesfromwhichclass-mediantra- levelsspacedsuchthatslantwisecirculationsofslope1:
jectoriesarecalculatedbytakingthemediantrajectory 40,withanabsoluteverticallevelspacingof300m,are
properties at each time in the coherent ensemble of resolvablenear600hPaandslopesshallowerthan1:50
trajectories.Trajectoryclassificationcanberepeatedfor areresolvablebelow750hPa.Thiscompareswellwith
trajectory populations calculated from start locations the1in50slopesuggestedasnecessarytosimulatethe
defined at consecutive times in the evolution of a release of conditional symmetric instability by slant-
weather system. Class-median trajectories from each wise circulations (Persson andWarner 1993) and sim-
consecutivetimecanbecalculatedrelativetotheposi- ilarly slantwise sting-jet descents (Clark et al. 2005;
tion of the center of the weather system at the given Grayetal.2011).
SEPTEMBER2015 HART ET AL. 3523
The simulation was initialized at 0000 UTC 8 De- conveyor belts can be distinguished by their high and
cemberfromtheoperationalNorthAtlanticandEuro- low u values, respectively [after Carlson (1980)]; and
e
peananalysis,withboundaryconditionsprovidedfrom Clark et al. (2005) show that u is also approximately
e
theoperational global MetUM. This modelingsetup is conservedduringthedescentofairinstingjets.
nearly identical to that used in MA14; the only differ-
a. Identificationofconveyorbelts
encesaretheupdatesduetothechangeinmodelversion
from7.3to8.2.SimulationsofFriedhelmfromthetwo To identify the conveyor belts, each trajectory was
model versions compare very closely (not shown). calculatedoverthetimeperiod[t 23,t 13h],wheret
0 0 0
Model data from the version 8.2 simulation were in- is the initialization time from when both forward and
terpolated onto pressure levels (Dp525hPa) before backward trajectories were calculated. This 7-h period
calculatingthediagnosticsusedinthisstudy. waschosenasaminimumtimespaninwhichtocapture
The aim is to demonstrate that the clustering meth- the key features of conveyor belt flows (i.e., location
odology can characterize airstreams that flow through relativetothestormcenter,curvature,andascendingor
thelow-leveljetregionsofthiscyclone.Inprinciple,the descending character). Tests with a longer time period
methodology could identify these low-level jets from [t 2 6, t 1 6h] produced very similar results (not
0 0
trajectories that describe the full flow around the cy- shown).AsWernliandDavies(1997)noted,these air-
clone. Jet regions of strong wind speed would be dis- streams are coherent for the duration of storm in-
tinguishedbycoherentclassesoftrajectoriesthattrace tensification, longer than 12h. However, these key
greater distances than those not associated with jet re- featuresarealsopresentonshortertimescales[e.g.,see
gions. To reduce the computational resources needed, Fig.10inSchultz(2001)].
including the computational challenge of clustering The resulting classification for Cyclone Friedhelm is
;105trajectories,wespeedupthisidentificationprocess shown for trajectories passingthrough low-level jet re-
by preselecting grid points within strong wind regions. gions at 0600 UTC 8 December 2011 (Fig. 3). The full
However, this preselection is a practical step, not a population of trajectories is shown in Fig. 3a. Classes
necessary one. To this end we choose where to seed with both CCB (class 1) and WCB (class 5) character-
trajectories by identifying grid points that lie within a istics are identified (Figs. 3b and 3c, respectively). At
threshold isotach in the lower troposphere (950– this time the cyclone structure was identified as corre-
650hPa).Considering25-hPapressureincrements,tra- spondingtostagethreeoftheShapiro–Keyserconcep-
jectoriesidentifyingtheconveyorbeltsateachlevelare tual model (Shapiro and Keyser 1990) by MA14: the
started within any 40ms21 isotach that is vertically fronts had formed a T-bone structure with the WCB
contiguous with a 40ms21 isotach at 850hPa, an arbi- still present and the CCB starting to wrap around the
trary but reasonable wind speed for low-level jets (re- cyclone center. Figure 3d presents a WCB trajectory
sultswillbeshowninsection3a).Forcomparisonwith population obtained by thresholding for saturated as-
thebent-backfrontjetsstudiedinMA14,theirthreshold cent. Because of the short 21-h duration of these
of 45ms21 is chosen (results will be shown in section trajectories, a 600-hPa ascent criterion [as applied in
3b). Start points were thus selected hourly from the previousstudiessuchasWernliandDavies(1997)fora
model output and used to initialize both forward and 48-hperiod]onlyadmits183trajectories.Sotoprovidea
backwardtrajectories.Trajectorieswerecalculatedwith comparablepopulationsizeforcomparisontotheWCB
the Lagrangian Analysis Tool (LAGRANTO; Wernli cluster class, a 400-hPa ascent criterion is chosen. The
and Davies 1997; Sprenger and Wernli 2015) using the WCBs obtained by the agglomerative clustering and
iterativeEulerschemeappliedtohourlymodeloutput, thresholdingmethodsmatchclosely(cf.Figs.3cand3d).
withaniterationtimestepof5min. Obtaining the WCB by thresholding was almost in-
The observation vector (W) for each trajectory is stantaneous, whereas cluster analysis of the full 9738
j
described by latitude, longitude, and pressure co- trajectoriespassingthroughthelow-leveljetregiontook
ordinates with equivalent potential temperature (u ) ;10min. (However, the clustering of the 887 trajecto-
e
providingtheairmasscharacteristic.Formoistflowsu riespassingthroughthestrongwindregionatthetipof
e
isaconservedvariable;however,u canevolveintime thebent-backfronttoproduceFig.7completesinorder
e
alongthetrajectoriesandtrajectorieswithsimilaru will seconds;thisissuewillbediscussedinsection4.)Median
e
bepreferentiallyclustered.Whileotherairmasstracers trajectoriesfortheCCBandWCBclasspopulationsare
couldbechosen,aprioriknowledgeofflowaroundex- overlaid in Figs. 3b and 3c. These median trajectories,
tratropical cyclones suggests inclusion of a measure of along with the others resulting from the clustering for
themoistentropyofairparcelsisworthwhile:Browning this start time, are mapped in Fig. 4a with Figs. 4b, 4c,
and Roberts (1994) describe how the warm and cold and 4d displaying their evolution in pressure, relative
3524 MONTHLY WEATHER REVIEW VOLUME143
FIG.3.(a)Fullpopulationoftrajectoriespassingthroughlow-leveljet(windspeedexceeding40ms21)at0600UTC8Dec2011.
(b)Trajectoriesassignedtoclusterclass1(CCB)withtheclassmedianoverlaid(shadedandblack-edgedthickline).(c)Asin(b),butfor
clusterclass5(WCB).(d)AWCBtrajectorypopulationobtainedfromthresholdingforsaturated(RH.90%)ascent(Dp. 400hPain
21h).Every10thtrajectoryisplottedin(a)witheverythirdshownin(b)–(d).Dotsindicate0600UTCinitiallocationsofdisplayed
trajectories,withinsetsshowingverticalcrosssectionsofthesestartlocationsbetween600and1000hPathroughthelongitudedrawnin
bold. Trajectories are clustered over the period 0300–0900 UTC, but trajectory histories are shown for the extended period 0100–
2200UTC.
humidity,andu ,respectively.Theseareshownforthe the CCB flow is captured in class 1. This median tra-
e
full time period of the simulation but, following the jectory remains saturated and below 800hPa while a
previous discussion, classification was only performed drierairstream,class2,followsamorezonalpathabove
on trajectory histories from 0300 to 0900 UTC. The 700hPa (Figs. 4b and 4c). Together with class 4, these
dendrogram associated with these clusters classes is three classes represent flow in the cool sector of the
found in Fig. 2a and shows the distance cutoff used to cyclone,northwestofthecoldfront,asdemonstratedin
obtainthesesixclasses. Fig.4d.Thewindmaximumsomedistancesoutheastof
The sensitivity of the final classification to variables thecyclonecenterinFig.4aisassociatedwiththecold
chosenforclusteringwastested(notshown)byconsider- front and attendant WCB. The classification in Fig. 4
ingboththenumberofclassesproducedandcomparison capturestheWCBwithclass5,asidentifiedbyamedian
of resulting class-median trajectories. This showed that trajectorythatstartsnearthesurfaceandrises100hPa
pressureevolutionwasastronglydistinguishingvariable, between 0300 and 0900 UTC before ascending more
withlatitudeandlongitudelessso.Inclusionofu inthe rapidly to about 350hPa (Fig. 4b). Class 6 describes a
e
observationvectortendedtoincreasethenumberoffinal similarairstreamatanelevatedaltitude.Thesemedian
clusters.Thisimpliesitassistsindistinguishingdifferentair trajectories remainsaturated,alongwithclass 3,which
massesconstitutingtheairstreams. captures flow through a lower extension of the upper-
The wind maximum directly south of and closest to leveljet.Classes3,5,and6summarizeflowinthewarm
the cyclone center is typical of the wraparound of the sectorofthiscyclone(Fig.4d).
CCBandtheassociateddevelopmentofstrongsurface Howcoherentaretheairstreamssummarizedbythese
winds (Fig. 4a). The near-surface airstream describing class medians? Following from Wernli and Davies
SEPTEMBER2015 HART ET AL. 3525
FIG.4.Classificationoftrajectoriesarrivinginlow-leveljets(windspeedexceeding40ms21)at0600UTC8Dec2011labeledineach
panelbyclassnumber.(a)Class-mediantrajectoriescoloredbypressurewiththepositionoftheminimumcyclonepressuremarkedby3
at0600UTConthesmoothedcyclonetrack(blackline).Thedirectionofthetrajectoriescanbedeterminedfromthenumericaltrajectory
labelsthatarepositionednearthebeginningofthetrajectories(att 23h).Contoursof40and45ms21windspeedaremarkedingray.
0
Class-medianevolutionsof(b)pressure,(c)relativehumidity,and(d)uewiththestartandendtimesofthe[t023,t013h]classification
perioddenotedbysquaresandstars,respectively.
(1997)whousedvariance,wequantifycoherencybythe at 0600 UTC in the evolution of Friedhelm. We now
standard deviations of the trajectory observation vari- considerthelaterhoursofdevelopment.
ables at each time and then average this value for all Agglomerative clustering of trajectories was applied
times (Table 1). Considering u , trajectory class 5 (the separately to the model output for initial times set to
e
WCB airstream) is the most coherent relative to other every hour of the simulation from 0500 to 1700 UTC.
classes.Likewiseclass1,theCCBhassmallvariancein Thisresultedinapopulationofclass-mediantrajectories
u . However, small standard deviations in pressure (i.e.,theinformationshowninFig.4foreachtime).The
e
(;50hPa, about 4 times the model vertical level spac- entire set (for all initial times) of class-median trajec-
ing)andu (;1K)inalltheclassessupporttheassertion tories can be classified as follows. Figure 5a shows an
e
that this clustering methodology is capturing coherent illustrativeselectionthoseclass-mediantrajectorieswith
ensemblesoftrajectories.Theseresultsprovideachar- WCB characteristics. Before classification the system-
acterizationofairstreamsconstitutingthelow-leveljets relative (relative to the storm center) class-median
TABLE1.Averagestandarddeviation(s)ofeachvariablefortheensembleoftrajectoriesineachclassobtainedfromclustering
trajectoriespassingthroughlow-levelwindmaximaat0600UTC(showninFig.4)and1600UTC(showninFig.7).Averagesforeach
variableiscomputedasthemeanofssateachhourfortheclassificationperiodchosen:[t 23,t 13h]at0600UTCand[t 25,t h]at
0 0 0 0
1600UTC.
0600UTCclasses 1600UTCclasses
1 2 3 4 5 6 1 2 3 4 5
s(lon;8) 0.53 0.62 0.98 1.43 0.52 0.93 0.35 0.22 0.24 0.31 0.27
s(lat;8) 0.70 0.80 0.34 0.63 0.71 0.80 0.27 0.16 0.25 0.22 0.21
s(pressure;hPa) 52.99 45.21 39.78 32.20 43.14 46.40 33.66 39.88 49.09 35.17 47.11
s(ue;K) 0.50 1.12 0.90 1.13 0.40 1.18 0.31 0.24 0.30 0.21 0.39
No.oftrajectories 1192 1847 1294 1742 1324 2339 249 89 188 133 228