Contraction Rate and Its Relationship to Frontogenesis, the Lyapunov Exponent, Fluid Trapping ...

17 Pages·2005·1.02 MB·English

MAY2005 COHEN AND SCHULTZ 1353
Contraction Rate and Its Relationship to Frontogenesis, the Lyapunov Exponent,
Fluid Trapping, and Airstream Boundaries
ROBERT A. COHEN
DepartmentofPhysics,EastStroudsburgUniversity,EastStroudsburg,Pennsylvania
DAVID M. SCHULTZ
CooperativeInstituteforMesoscaleMeteorologicalStudies,UniversityofOklahoma,andNOAA/NationalSevereStormsLaboratory,
Norman,Oklahoma
(Manuscriptreceived2June2004,infinalform25October2004)
ABSTRACT
AlthoughakinematicframeworkfordiagnosingfrontogenesisexistsintheformofthePetterssenfron-
togenesisfunctionanditsvectorgeneralization,asimilarframeworkfordiagnosingairstreamboundaries
(e.g.,drylines,leetroughs)hasnotbeenconstructed.Thispaperpresentssuchaframework,beginningwith
akinematicexpressionfortherateofchangeoftheseparationvectorbetweentwoadjacentairparcels.The
maximumgrowthrateoftheseparationvectoriscalledtheinstantaneousdilatationrateanditsorientation
iscalledtheaxisofdilatation.Similarly,amaximumdecayrateiscalledtheinstantaneouscontractionrate
anditsorientationiscalledtheaxisofcontraction.Theseexpressionsarerelatedtothevectorfrontogenesis
function,inthatthegrowthrateoftheseparationvectorcorrespondswiththescalarfrontogenesisfunction,
and the rotation rate of the separation vector corresponds with the rotational component of the vector
frontogenesisfunction.
Becausevorticitycanrotateair-parcelpairsoutofregionsofdeformation,theinstantaneousdilatation
andcontractionratesandaxesmaynotbeappropriatediagnosticsofairstreamboundariesforfluidflows
ingeneral.Rather,thegrowthrateandorientationofanairstreamboundarymaycorrespondbettertothe
so-calledasymptoticcontractionrateandtheasymptoticaxisofdilatation,respectively.Expressionsforthe
asymptotic dilatation and contraction rates, as well as their orientations, the asymptotic dilatation and
contractionaxes,arederived.TheasymptoticdilatationrateisrelatedtotheLyapunovexponentforthe
flow.Inaddition,afluid-trappingdiagnosticisderivedtodistinguishamongadjacentparcelsbeingpulled
apart, being pushed together, or trapped in an eddy. Finally, these diagnostics are applied to simple,
idealized,steady-stateflowsandanonsteadyidealizedvortexinnondivergent,diffluentflowtoshowtheir
utilityfordeterminingthecharacterofair-parceltrajectoriesandairstreamboundaries.
1. Introduction speed and direction, but, unlike fronts, are not neces-
sarilyassociatedwith,orcharacterizedby,temperature
Interesting weather in the atmosphere often occurs gradients[seethediscussioninsection2bofCohenand
along boundaries between different air masses. Fronts Kreitzberg (1997) and in section 4 of Sanders (1999)].
areoneexampleofsuchboundaries,asflowswithcon- We term these features airstream boundaries because
vergenceanddeformationconcentrateisothermsatthe airstreams having different origins are brought closer
edges of air masses. In conjunction with this frontoge- together to form boundaries in the flow.
nesis, secondary horizontal and vertical ageostrophic Akinematicframeworktodiagnosefrontogenesisin
circulations result to restore thermal-wind balance, as a Lagrangian sense (following air parcels) has a long
pergeostrophicandsemigeostrophictheory.Otherex- history(e.g.,Petterssen1936;Miller1948;Keyseretal.
amplesofboundariesincludedrylines,leetroughs,and 1988;Keyser1999).Incontrast,whereasframeworksto
baroclinictroughs(Sanders1999),featuresthatarefun- studythecoherenceofairstreamshavebeendeveloped
damentally characterized by abrupt changes in wind (e.g., Carlson 1980; Browning 1986, 1990; Wernli and
Davies 1997; Wernli 1997), quantitative approaches to
diagnose the boundaries between airstreams have not
been as extensively studied. One approach is that of
Correspondingauthoraddress:Dr.DavidM.Schultz,NOAA/
CohenandKreitzberg(1997).Theydefinetheaverage
NationalSevereStormsLaboratory,1313HalleyCircle,Norman,
contraction rate C as the time-averaged rate at which
OK73069.
E-mail:[email protected] the three-dimensional distance between two adjacent
©2005AmericanMeteorologicalSociety
MWR2922
1354 MONTHLY WEATHER REVIEW VOLUME133
air-parcel changes over some previous finite time pe- sion for what we term the fluid-trapping diagnostic.
riod t ending at t (cid:1) 0: This diagnostic is a measure of whether adjacent air
(cid:1) (cid:2) parcelsapproacheachother,separatefromeachother,
1 r(cid:3)t(cid:4) or are trapped together in an eddy. Such a diagnostic
C(cid:1)(cid:2) ln ; t(cid:1)0, (cid:3)1(cid:4)
t r(cid:3)0(cid:4) canbeusedtoassesswhethertheflowischaracterized
by straining or eddies. In a straining-dominated flow,
where r(t) is the distance between the two parcels at adjacent air parcels approach or separate from each
time t. Cohen and Kreitzberg (1997) calculate C by other.Incontrast,inaneddy-dominatedflow,thesepa-
computingair-parceltrajectoriesfrommesoscalemodel ration vector between adjacent air parcels oscillates in
outputovertheentiremodeldomainbackintime12h. lengthandrotates,trappingthefluidinsidetheeddies.
Thedistancesbetweenadjacenttrajectoriesattheirori- Thefluid-trappingdiagnostichaswiderapplicationsto
gins, r((cid:2)12 h), and their final locations, r(0), can then turbulence, chaos, and predictability (e.g., McWilliams
beusedtocalculateCasexpressedin(1).Asshownin 1984;Benzietal.1988;PierrehumbertandYang1993).
CohenandKreitzberg(1997),regionsoflargepositive The utility of these expressions is illustrated in section
12-h average contraction rate are associated with air- 4bytheirapplicationtoanidealized,nondivergentflow
stream boundaries in a rapidly developing midlatitude resembling an extratropical cyclone in a diffluent jet-
cyclone.Thus,theaveragecontractionratecanbeused exitregion,whereregionsoflargeasymptoticcontrac-
as a quantitative measure of the strength of an air- tion rate correspond to airstream boundaries. The ar-
stream boundary. ticle concludes in section 5 with a summary and some
Such a diagnosis of average contraction rates, how- possible applications of these kinematic expressions to
ever, requires computing fluid-parcel trajectories, a observed atmospheric flows. Appendix A is a list of
computationally expensive process requiring special- symbolsfoundinthispaper.Aderivationoftheinstan-
ized code. In addition, high temporal and spatial reso- taneouslocalLyapunovexponentdiscussedinsection3
lution may be needed, especially for diagnosing small- is presented in appendix B.
scale features. If, however, a kinematic expression for
thecontractionratewerepossible,thedevelopmentof
airstream boundaries could be assessed without the 2. Instantaneous dilatation and contraction rates
temporal history needed to compute airstream bound-
Considertwofluidparcelsseparatedbyavectordis-
aries using the approach of Cohen and Kreitzberg
tance(cid:5)r(t),theinfinitesimalvectoranalogtor(t)in(1).
(1997).
Owingtogradientsoffluidmotionwithintheflow,the
Thepurposeofthispaperistodevelopanexpression
velocityofthetwoparcelsmaybedifferent,leadingto
for the contraction rate using fundamental kinematic
achangeintheirvectorseparation(cid:5)r(t).If(cid:5)r(t)changes
quantities.Insection2,expressionsarederivedforthe
quickly,trajectoriesofthefluidparcelswouldbehighly
instantaneous dilatation and contraction rates. These
dependent on their initial positions. Such sensitive de-
kinematicexpressionsarethemaximumandminimum
pendenceofparceltrajectoriesontheirinitialpositions
ratesofchangeofthedistancebetweentwoadjacentair
ischaracteristicofchaoticflowandhasimplicationsfor
parcels,respectively,andtheinstantaneouscontraction
the predictability of the flow.
rate is the instantaneous analog to the time-averaged
Tosimplifyanalysisandinterpretation,onlyhorizon-
contraction rate C. In addition, the instantaneous con-
tal motions are considered. To represent the time de-
traction rate is related to the expression for accumula-
pendenceoftheseparationintwodimensionsbetween
tion(Saucier1955,section10.39)andthefrontogenesis
thesetwofluidparcels,theseparationvectorforasmall
function(Petterssen1936;Keyseretal.1988).Theaxis
periodoftime(cid:5)r(t)iswrittenintermsofagrowthrate
of dilatation represents the direction of maximum
(cid:6)and rotation rate (cid:7):
stretching of fluid parcels by the deformation in the
flow.Whenvorticityispresentintheflow,thevorticity
(cid:2)r(cid:3)t(cid:4)(cid:1)(cid:2)r(cid:3)0(cid:4)exp(cid:8)(cid:3)(cid:3)(cid:9)i(cid:4)(cid:4)t(cid:10), (cid:3)2(cid:4)
will rotate air-parcel pairs from the regions of maxi-
mum instantaneous dilatation and contraction rates. where the real and imaginary parts of (cid:5)r(t) can be
Consequently,expressionsforwhatwetermtheasymp- viewed as components of a vector ((cid:5)x, (cid:5)y):
toticdilatationandcontractionratesarederivedinsec-
tion 3. The directions associated with the asymptotic (cid:2)r(cid:3)t(cid:4)(cid:1)(cid:2)x(cid:9)i(cid:2)y. (cid:3)3(cid:4)
dilatation and contraction rates are called the asymp-
totic dilatation and contraction axes, respectively. Written in this way, the magnitude of the separation
Where the asymptotic contraction rate is positive, the vectorwouldincreaseexponentiallywithtimeatarate
asymptoticdilatationaxisrepresentstheorientationof (cid:6), and the separation vector would rotate with an an-
the instantaneous processes acting to create an air- gular frequency (cid:7), positive for counterclockwise rota-
streamboundary.Inaddition,theasymptoticdilatation tions.
rateisrelatedtotheinstantaneouslocalLyapunovex- These expressions can be solved for (cid:6)and (cid:7), result-
ponent. These derivations yield an intriguing expres- ing in the growth rate and rotation rate of the separa-
MAY2005 COHEN AND SCHULTZ 1355
(cid:5) (cid:5)
tion vector throughout the flow for small times. From d(cid:3) d(cid:3)
(cid:1) (cid:1)0.
(3), the rate of change of the separation vector is d(cid:7) d(cid:7)
(cid:7)(cid:1)(cid:9) (cid:7)(cid:1)(cid:9)
d c
d(cid:3)(cid:2)r(cid:4) d(cid:3)(cid:2)x(cid:4) d(cid:3)(cid:2)y(cid:4)
(cid:1) (cid:9)i . Solvingtheaboveequationleadstothefollowingrela-
dt dt dt tionship:
(cid:3) (cid:4)(cid:6)(cid:3) (cid:4)
ATaylorexpansionoverasmalltimeperiod,retaining (cid:5)(cid:6) (cid:5)u (cid:5)u (cid:5)(cid:6)
tan2(cid:9) (cid:1)tan2(cid:9) (cid:1) (cid:9) (cid:2)
only the linear terms, yields d c (cid:5)x (cid:5)y (cid:5)x (cid:5)y
(cid:3) (cid:4) (cid:3) (cid:4)
d(cid:3)(cid:2)r(cid:4) (cid:5)u (cid:5)u (cid:5)(cid:6) (cid:5)(cid:6) (cid:1)E (cid:10)E . (cid:3)6(cid:4)
(cid:1) (cid:2)x(cid:9) (cid:2)y (cid:9)i (cid:2)x(cid:9) (cid:2)y . (cid:3)4(cid:4) sh st
dt (cid:5)x (cid:5)y (cid:5)x (cid:5)y Theorientation(cid:15) correspondstotheaxisofdilatation,
d
This is equivalent to the linear strain rate of a fluid the orientation of maximum growth rate. The value of
element in each direction (e.g., Kundu 1990, 54–55). this maximum growth rate (cid:6)|(cid:11)(cid:1)(cid:15) is termed d, the in-
Doswell (1984) showed that the neighborhood of a stantaneousdilatationrate.Theothderorientation(cid:15)ccor-
point within which neglect of higher-order terms is responds to the axis of contraction, the orientation at
valid is approximately one-tenth the horizontal scaling whichthegrowthrateisaminimum.Correspondingly,
length. the instantaneous contraction rate c is defined as the
Substituting (2) and (3) into (4) results in negative of the minimum growth rate (i.e., the maxi-
(cid:3) (cid:4) mumdecayrate(cid:2)(cid:6)| ).Theinstantaneouscontrac-
(cid:11)(cid:1)(cid:15)
(cid:5)u (cid:5)u tion rate c is the instantacneous analog to the average
(cid:3)(cid:3)(cid:9)i(cid:4)(cid:4)(cid:3)(cid:2)x(cid:9)i(cid:2)y(cid:4)(cid:1) (cid:2)x(cid:9) (cid:2)y
(cid:5)x (cid:5)y contractionrateCin(1).Notethatthetwosolutionsto
(cid:3) (cid:4) (6) must be orthogonal (i.e., (cid:15) (cid:1) (cid:15) (cid:9) (cid:16)/2), because
(cid:5)(cid:6) (cid:5)(cid:6) c d
(cid:9)i (cid:2)x(cid:9) (cid:2)y . tan2(cid:11)(cid:1) tan2((cid:11)(cid:9) (cid:16)/2). Thus, one orientation can be
(cid:5)x (cid:5)y obtained via
(cid:3) (cid:4)
Theexpressionhastwoparts,arealpartandanimagi- 1 E
nary part, resulting in two independent equations: (cid:9)(cid:1) tan(cid:2)1 sh , (cid:3)7(cid:4)
2 E
st
(cid:5)u (cid:5)u andtheotherorientationcanbeobtainedbyadding(cid:16)/2
(cid:3)(cid:2)x(cid:2)(cid:4)(cid:2)y(cid:1) (cid:2)x(cid:9) (cid:2)y,
(cid:5)x (cid:5)y to (7).
To rewrite the expressions for the growth and rota-
(cid:5)(cid:6) (cid:5)(cid:6)
(cid:3)(cid:2)y(cid:9)(cid:4)(cid:2)x(cid:1)(cid:5)x(cid:2)x(cid:9)(cid:5)y(cid:2)y. tion rates (5a) and (5b) in terms of (cid:15)(i.e., (cid:15)d and (cid:15)c),
(6) can be expressed as
Substituting(cid:5)x(cid:1)|(cid:5)r|cos(cid:11)and(cid:5)y(cid:1)|(cid:5)r|sin(cid:11),where(cid:11) sin2(cid:9)(cid:1)(cid:17)E (cid:10)E,
sh
is the orientation of the parcel pair, yields cos2(cid:9)(cid:1)(cid:17)E (cid:10)E,
(cid:3) (cid:4) st
(cid:3)(cid:1)(cid:5)ucos2(cid:7)(cid:9)(cid:5)(cid:6)sin2(cid:7)(cid:9) (cid:5)u(cid:9)(cid:5)(cid:6) sin(cid:7)cos(cid:7), wheretheresultantdeformationE(cid:1)(E2st(cid:9)E2sh)1/2,and
(cid:5)x (cid:5)y (cid:5)y (cid:5)x the two signs on the right-hand side correspond to the
(cid:3) (cid:4)
(cid:5)(cid:6) (cid:5)u (cid:5)(cid:6) (cid:5)u twovaluesof(cid:15).Ascanbeshownbytakingthesecond
(cid:4)(cid:1) cos2(cid:7)(cid:2) sin2(cid:7)(cid:9) (cid:2) sin(cid:7)cos(cid:7). derivative of (cid:6) with respect to (cid:11), the positive value
(cid:5)x (cid:5)y (cid:5)y (cid:5)x corresponds to (cid:15) and the negative value corresponds
d
Using trigonometric relationships, these expressions to (cid:15)c. Using these expressions in (5a) and (5b), the
can be simplified to deformation terms Esh and Est, which depend on the
coordinate system, are replaced by the preferred Gal-
1 ilean-invariant resultant deformation E (hereafter, re-
(cid:3)(cid:1) (cid:3)E sin2(cid:7)(cid:9)E cos2(cid:7)(cid:9)D(cid:4), (cid:3)5a(cid:4)
2 sh st ferred to simply as deformation):
(cid:4)(cid:1)1(cid:3)E cos2(cid:7)(cid:2)E sin2(cid:7)(cid:9)(cid:8)(cid:4), (cid:3)5b(cid:4) (cid:3)(cid:1)1(cid:3)(cid:17)Esin2(cid:9)sin2(cid:7)(cid:17)Ecos2(cid:9)cos2(cid:7)(cid:9)D(cid:4),
2 sh st 2
where divergence D (cid:1) (cid:12)u/(cid:12)x (cid:9) (cid:12)(cid:13)/(cid:12)y, relative vorticity (cid:4)(cid:1)1(cid:3)(cid:17)Esin2(cid:9)cos2(cid:7)(cid:18)Ecos2(cid:9)sin2(cid:7)(cid:9)(cid:8)(cid:4).
(cid:14)(cid:1) (cid:12)(cid:13)/(cid:12)x (cid:2) (cid:12)u/(cid:12)y (hereafter referred to as vorticity), 2
stretching deformation E (cid:1) (cid:12)u/(cid:12)x (cid:2) (cid:12)(cid:13)/(cid:12)y, and shear-
st Employing trigonometric identities yields
ing deformation E (cid:1) (cid:12)u/(cid:12)y (cid:9) (cid:12)(cid:13)/(cid:12)x.
sh
Ingeneral,ateachpointintheflow,anorientation(cid:15) 1
exists for which the growth rate (cid:6) is largest, and and (cid:3)(cid:1)2(cid:8)(cid:17)Ecos2(cid:3)(cid:9)(cid:2)(cid:7)(cid:4)(cid:9)D(cid:10), (cid:3)8a(cid:4)
orientation(cid:15) existsforwhichthegrowthrateissmall-
c
est. These maximum and minimum values of (cid:6)occur (cid:4)(cid:1)1(cid:8)(cid:17)Esin2(cid:3)(cid:9)(cid:2)(cid:7)(cid:4)(cid:9)(cid:8)(cid:10). (cid:3)8b(cid:4)
where 2
1356 MONTHLY WEATHER REVIEW VOLUME133
(cid:3) (cid:4) (cid:3) (cid:4)
Itcanbeshownthat(cid:6)in(8a)equalsthenegativeofthe (cid:5)(cid:6) (cid:5)(cid:6)
(cid:6)(cid:1)(cid:6) (cid:9) x(cid:9) y
partial derivative along an arbitrarily chosen direction 0 (cid:5)x (cid:5)y
oftheprojectionofvectorwindvelocityontothesame
direction.Asimilarrelationshipappearsintheexpres- 1
(cid:1) (cid:3)(cid:6) (cid:9)Dy(cid:9)E x(cid:2)E y(cid:9)(cid:8)x(cid:4), (cid:3)13b(cid:4)
sions for the frontogenesis function, as discussed later 2 0 sh st
in section 2b. where (u , (cid:13)) represents a uniform translation. Here,
When (cid:11)(cid:1) (cid:15), (cid:7)(cid:1) (cid:14), implying that, for parcel pairs 0 0
weassumethatthederivativesofthewindfield,aswell
situatedalongtheaxesofdilatationorcontraction,vor- as u , (cid:13), D, E , E , and (cid:14), are constant in space and
ticity is the only process by which rotation of these 0 0 sh st
time,followingtheassumptionsfromtheTaylorexpan-
parcelpairsoccurs.Inaddition,when(cid:11)(cid:1)(cid:15),(cid:6)iseither
sionin(4).Similaridealizedflowstructureswereillus-
a maximum or minimum with the values:
tratedinPetterssen(1936;1940,chapter5;1956,section
(cid:17)E(cid:9)D 2.7). Figure 1a shows the above flow field with a pure
(cid:3)|(cid:7)(cid:1)(cid:9)(cid:1) 2 , (cid:3)9(cid:4) uniformdeformationE:Est(cid:1)E(u0(cid:1)(cid:13)0(cid:1)Esh(cid:1)D(cid:1)
(cid:14)(cid:1) 0).
respectively. The right-hand side of (9) is maximum
The frontogenetic aspects of this flow were first dis-
whenthesignispositiveandminimumwhenthesignis
cussed by Bergeron (1928). Both the instantaneous di-
negative,inagreementwiththemeaningidentifiedear-
latationandcontractionratesareuniformoverthedo-
lier. Because the instantaneous dilatation rate is de-
main and are equal to E/2 by (10a) and (10b), respec-
finedasthemaximumgrowthrated,andtheinstanta-
tively.Becauseofthelinearnatureoftheflowin(13a)
neouscontractionrateisdefinedasthenegativeofthe and(13b),(cid:15) and(cid:15) havethesameorientationatevery
minimum growth rate c, d c
point in the flow, although this is not the case in non-
E(cid:9)D linearflows.Theaxesofdilatationandcontractionare
d(cid:1)(cid:3)| (cid:1) , (cid:3)10a(cid:4)
(cid:7)(cid:1)(cid:9)d 2 always perpendicular to each other in this, and every,
flow(e.g.,Fig.1).Whendivergenceequalandopposite
and
tothedeformationisaddedtotheflow(Fig.1b),c(cid:1)E
E(cid:2)D and d (cid:1) 0, such that the flow is characterized by pure
c(cid:1)(cid:2)(cid:3)| (cid:1) . (cid:3)10b(cid:4)
(cid:7)(cid:1)(cid:9)c 2 contraction and no dilatation. The airstream bound-
aries in the two flows in Fig. 1 are oriented east–west,
To simplify notation, the expressions in (8a) and (8b)
parallel to the axis of dilatation.
for(cid:6)and(cid:7),respectively,canbewrittenintermsof(cid:15):
d
1 b. Relationship to accumulation and the vector
(cid:3)(cid:1)2(cid:8)Ecos2(cid:3)(cid:9)d(cid:2)(cid:7)(cid:4)(cid:9)D(cid:10), (cid:3)11a(cid:4) frontogenesis function
1 Because(cid:6)and(cid:7)reflecttheinstantaneousmotionof
(cid:4)(cid:1) (cid:8)Esin2(cid:3)(cid:9) (cid:2)(cid:7)(cid:4)(cid:9)(cid:8)(cid:10). (cid:3)11b(cid:4) anyparcelpair(i.e.,changesinthemagnitudeandori-
2 d
entation of the corresponding separation vector), the
Themaximumcounterclockwise(positive)orclockwise time tendency of conserved passive tracers can be di-
(negative)rotationrateoccurs45°betweentheaxesof agnosed using these expressions. For example, given a
dilatation and contraction. Using (11b), these maxi- conserved quantity Q,
mum and minimum rotation rates, (cid:19) and (cid:19) , can be
(cid:9) (cid:2) (cid:2)Q
determined: |(cid:12)Q|(cid:1) ,
(cid:2)r
(cid:17)E(cid:9)(cid:8)
(cid:11)(cid:17)(cid:1) 2 . (cid:3)12(cid:4) where (cid:20) (cid:1) i (cid:12)/(cid:12)x (cid:9) j (cid:12)/(cid:12)y, and (cid:5)r (cid:1) |(cid:5)r|, the distance
separatingtwoparcelsorientedalongthegradient.The
This equation shows that deformation increases the
timetendencyofthemagnitudeofthegradientcanbe
maximumrotationrateoftheparcelpairoverthatdue
written
tovorticityalone.Likewise,deformationdecreasesthe (cid:3) (cid:4)
d|(cid:12)Q| d (cid:2)Q
minimumrotationrateoftheparcelpairoverthatdue (cid:1) ,
to vorticity alone. dt dt (cid:2)r
a. Examples d|(cid:12)Q| (cid:2)Q d(cid:3)(cid:2)r(cid:4)
(cid:1)(cid:2) . (cid:3)14(cid:4)
dt (cid:3)(cid:2)r(cid:4)2 dt
To illustrate these expressions for some idealized
flow patterns, consider a flow field: From (2),
(cid:3) (cid:4) (cid:3) (cid:4)
(cid:5)u (cid:5)u (cid:2)r(cid:1)(cid:2)r(cid:3)0(cid:4)exp(cid:3)(cid:3)| t(cid:4),
u(cid:1)u (cid:9) x(cid:9) y (cid:7)(cid:1)(cid:13)(cid:9)(cid:14)(cid:10)2
0 (cid:5)x (cid:5)y or
1 1 d(cid:3)(cid:2)r(cid:4)
(cid:1) (cid:3)u (cid:9)Dx(cid:9)E y(cid:9)E x(cid:2)(cid:8)y(cid:4), (cid:3)13a(cid:4) (cid:3)| (cid:1) ,
2 0 sh st (cid:7)(cid:1)(cid:13)(cid:9)(cid:14)(cid:10)2 (cid:2)r dt
MAY2005 COHEN AND SCHULTZ 1357
FIG.1.Idealizedwindfieldsgivenby(13a)and(13b)inthetext.Here,(cid:14)(cid:1)0.Solidlinesindicatetheaxesofdilatation((cid:15)d)and
contraction((cid:15)).(a)Purestretchingdeformation:E (cid:1)E;u (cid:1)(cid:13) (cid:1)E (cid:1)D(cid:1)0.(b)Asin(a),exceptD(cid:1)(cid:2)E.
c st 0 0 sh
where d
(cid:3) (cid:4) Fn(cid:1)(cid:2)dt|(cid:12)(cid:16)|,
(cid:5)Q(cid:10)(cid:5)x
(cid:13)(cid:1)tan(cid:2)1 (cid:2)
(cid:5)Q(cid:10)(cid:5)y 1
F (cid:1)(cid:2) |(cid:12)(cid:16)|(cid:3)Ecos2(cid:15)(cid:2)D(cid:4). (cid:3)16a(cid:4)
n 2
is the local orientation of a line of constant Q and is
perpendicular to the orientation of the separation vec- Similarly,bymultiplying|(cid:20)(cid:23)|bytherotationrate(cid:7)|(cid:11)(cid:1)(cid:21)
tor (cid:11)(i.e., (cid:11)(cid:1) (cid:21)(cid:9) (cid:16)/2). This expression for (cid:6)repre- (i.e., the rotation rate valid for a parcel pair oriented
sents the linear strain rate in the direction of the gra- alongtheisentropeor,moresimply,therotationrateof
dient. Thus, (14) can be written theisentrope)in(11b),theexpressionfortherotational
component F of the vector frontogenesis function,
s
d|(cid:12)Q| (2.10b) in Keyser et al. (1988), can be derived:
(cid:1)(cid:2)|(cid:12)Q|(cid:3)| .
dt (cid:7)(cid:1)(cid:13)(cid:9)(cid:14)(cid:10)2
1
F (cid:1) |(cid:12)(cid:16)|(cid:3)Esin2(cid:15)(cid:9)(cid:8)(cid:4). (cid:3)16b(cid:4)
Substituting (11a) yields s 2
d|(cid:12)Q| 1 AsdiscussedinmoredetailbyKeyseretal.(1988),the
(cid:1)(cid:2) |(cid:12)Q|(cid:8)Ecos2(cid:3)(cid:9) (cid:2)(cid:13)(cid:2)(cid:14)(cid:10)2(cid:4)(cid:9)D(cid:10). expression for F is equal and opposite to the
dt 2 d n
LagrangianfrontogenesisfunctionFdevelopedbyPet-
Defining (cid:22)(cid:1) (cid:15) (cid:2) (cid:21), terssen(1936).Equation(16a)expressestheresultthat,
d
iftheanglebetweentheaxisofdilatationandtheisen-
d|(cid:12)Q| 1 trope is less than 45°, deformation is frontogenetical.
(cid:1) |(cid:12)Q|(cid:3)Ecos2(cid:15)(cid:2)D(cid:4). (cid:3)15(cid:4) Convergence is frontogenetical irrespective of the ori-
dt 2
entation of the isentropes. Diagnosis of F for a given
n
This expression is termed accumulation A(Q) (e.g., flow field indicates the forcing for vertical motion on
Saucier1955,363–373).Forexample,ifQisthepoten- the frontal scale.
tial temperature (cid:23), (15) becomes Equation(16b),ontheotherhand,expressesamea-
sure of the rotation of the isentropes, where positive
d valuesofF implycyclonicrotation.Forexample,ifthe
A(cid:3)(cid:16)(cid:4)(cid:1) |(cid:12)(cid:16)|, s
dt axis of dilatation is oriented between 0° and 90° coun-
terclockwiseoftheisentropes,thethermalgradientro-
whichisequivalenttothePetterssen(1936)frontogen- tates counterclockwise. In the Northern Hemisphere,
esisfunctionF.Asshownby(2.3a)and(2.10a)inKey- cyclonic vorticity results in the counterclockwise rota-
seretal.(1988),Fisrelatedtoexpressionsforthescalar tion of the thermal gradient, whereas anticyclonic vor-
frontogenesis function F of the vector frontogenesis: ticity results in the clockwise rotation of the thermal
n
1358 MONTHLY WEATHER REVIEW VOLUME133
gradient, both irrespective of the orientation of the growthanddecay,respectively,oftheseparationvector
isentropes. Diagnosis of F for a given flow field indi- between adjacent parcels, in general, parcels oriented
s
catestheforcingforverticalmotiononthescaleofthe along these axes experience such rates of growth and
baroclinic wave. Keyser et al. (1988) also showed the decay (d and c, respectively) instantaneously only.
relationship between the vector form of the fronto- Thesediagnosticscannotexpressthelong-termforma-
genesisfunctionandtheQvector(Hoskinsetal.1978). tion and evolution of airstream boundaries, much as
Barnes and Colman (1993, 1994), for example, have Petterssen (1936) frontogenesis cannot express the
applied these concepts to observed weather events. long-term formation and evolution of fronts. The goal
Tosummarizethissection,relationshipsweredevel- of this section is to derive expressions to describe the
oped based on the rate of change of the separation long-term behavior of air-parcel pairs.
vector,theinfinitesimalvectorconnectingtwoadjacent Consideraflowwithpuredeformation(Fig.1a)ora
air parcels. The instantaneous dilatation and contrac- flowwithdeformationanddivergence(Fig.1b).Aswas
tion rates d and c were introduced, and their relation- shown in section 2a, the axis of dilatation and the air-
ship to axes of dilatation and contraction, (cid:15) and (cid:15), stream boundary have the same orientation in such
d c
respectively, were detailed. The maximum and mini- flows. If a parcel pair is oriented along the axis of di-
mumrotationratesforadjacentairparcels,(cid:19) and(cid:19) , latation,itexperiencesthelargestinstantaneousdilata-
(cid:9) (cid:2)
respectively,werepresented.Theintimaterelationship tion rate for that location instantaneously. When vor-
betweenthegrowthrateoftheseparationvector(cid:6)and ticity is present, however, parcel pairs cannot stay in
scalar frontogenesis function F , and the rotation rate thatorientation,asdiscussedinregardto(8b);inother
n
of the separation vector (cid:7)and the rotational compo- words, vorticity rotates the air-parcel pair away from
nent of the vector frontogenesis function F, was dem- the axis of dilatation. In a similar way, the axis along
s
onstrated. In addition, all the above expressions are which the strongest deformation occurs (axis of dilata-
writtenintermsofGalileaninvariantquantities,imply- tion) is not the same axis along which the air parcels
ingthattheseexpressionsarenotdependentonapar- experiencelong-termseparation.Itfollowsthattheaxis
ticular reference frame. This framework identifies the of dilatation may not be the axis along which the air-
strength and orientation of airstream boundaries in streamboundaryforms,asisshownforfrontsinwhich
flowswithdeformationanddivergence.Whenvorticity theaxisofdilatationformsanacuteanglewiththefront
is present, however, more general expressions to diag- (Keyser et al. 1988; Bishop 1996; Schultz 2004).
nosetheairstreamboundariesmayberequired.These Becauserotationisresponsibleformovingair-parcel
are developed in the next section. pairs out of alignment with the axes of dilatation and
contraction, the instantaneous contraction rate reflects
a long-term growth rate only if the air-parcel pair is
3. Asymptotic dilatation and contraction rates aligned with an axis of zero rotation. Consequently,
(8b) is solved when (cid:7)(cid:1) 0:
Consider the Petterssen (1936) frontogenesis func-
(cid:8)(cid:1)(cid:17)Esin2(cid:3)(cid:9)(cid:2)(cid:17)(cid:4), (cid:3)17(cid:4)
tion F. The frontogenesis function quantitatively as-
sessestheinstantaneousprocessesthatleadtoachang- where(cid:11)hasbeenreplacedwith(cid:24),theorientationofthe
ing magnitude of the horizontal potential temperature
axis of nonrotation. In order for a unique axis of non-
gradientfromtheperspectiveofanairparcel.Inother rotationtoexist,(17)statesthatE(cid:18)|(cid:14)|.WhenE(cid:18)|(cid:14)|,
words,thePetterssen(1936)frontogenesisfunctionisa
the two solutions to (17) are
kinematicLagrangianexpression:theexistenceofare- (cid:3) (cid:4)
gionofpositiveFdoesnotnecessarilyimplythatafront 1 (cid:8)
eventually forms in that area. Because Petterssen (cid:17)d(cid:1)(cid:9)d(cid:9)2sin(cid:2)1 E , (cid:3)18a(cid:4)
(1936)frontogenesisisaninstantaneousLagrangiandi- (cid:3) (cid:4)
agnostic,airparcelsmusttravelwithinregionsoflarge 1 (cid:8)
frontogenesis to experience the prolonged effects of (cid:17)c(cid:1)(cid:9)c(cid:2)2sin(cid:2)1 E , (cid:3)18b(cid:4)
frontogenesis.AsnotedbyHoskinsandWest(1979,p.
1674), Hoskins (1982, p. 146), Xu (1988, p. 1234), and where(cid:24) and(cid:24) arecalledtheasymptoticdilatationaxis
d c
Rotunnoetal.(1994,p.3373),theformationofafront and asymptotic contraction axis, respectively. These
occurs where air parcels reside in a region of positive correspond to the axis of outflow and axis of inflow,
Petterssen (1936) frontogenesis for a sufficiently long respectively, of Petterssen (1936; 1940, 252–256; 1956,
time. Doswell (1984) showed that, although vorticity p.38).When|(cid:14)|(cid:25)E,thereisnosolutionwhere(cid:7)(cid:1)0,
couldnotcontributedirectlytoPetterssen(1936)fron- and no asymptotic dilatation or contraction axes exist.
togenesis,itcouldrotatetheisothermsintoapositionin When(cid:14)(cid:1)0,(cid:11)(cid:1)(cid:24),andtheaxisofdilatationandaxis
the flow that was more favorable frontogenetically. of contraction are equal to the asymptotic dilatation
The diagnostics in the last section are very much axisandasymptoticcontractionaxis,respectively(e.g.,
analogous to these aspects of frontogenesis. Although section 2a).
(cid:15) and (cid:15) represent the orientations of maximum Equation (17) can be rewritten
d c
MAY2005 COHEN AND SCHULTZ 1359
(cid:8)2 Finally, the orientation of (cid:24) is aligned along the di-
cos22(cid:3)(cid:9)(cid:2)(cid:17)(cid:4)(cid:1)1(cid:2) . (cid:3)19(cid:4) rection in which the separatiocn vector experiences its
E2
mostrapidrateofcontractioninthelongerterm,much
Squaring the expression for the growth rate (cid:6), (11a), as (cid:15)c represents the directions in which the separation
and employing (19) yields vector experiences its most rapid rate of decrease in-
stantaneously. Thus, if an airstream boundary were to
(cid:3)| (cid:1)1(cid:8)D(cid:17)(cid:3)E2(cid:2)(cid:8)2(cid:4)1(cid:10)2(cid:10). (cid:3)20(cid:4) form, (cid:24)d represents its likely orientation, given the as-
(cid:7)(cid:1)(cid:17) 2 sumptionsaboutsteady,linearflow.Becausesuchlong-
term or time-averaged contraction rates are related to
BecauseE(cid:18)|(cid:14)|foranaxisofnonrotationtoexist,(20)
airstreamboundaries(section1),theasymptoticdilata-
ismaximumwhenthesignfollowingDispositiveandis
tionaxisandasymptoticcontractionrateareproposed
minimum when the sign is negative. Consequently, (cid:24)
d tobebettermeasuresofthedevelopmentofairstream
and (cid:24) are defined such that (cid:6)| (cid:18) (cid:6)| .
c (cid:11)(cid:1)(cid:24)d (cid:11)(cid:1)(cid:24)c boundaries than the axis of dilatation and instanta-
Defining the asymptotic dilatation rate (D) and the
neouscontractionrate.Thisstatementisreminiscentof
asymptotic contraction rate (C) to be the growth rate
that in Petterssen (1940, p. 255): “the axis of outflow
along (cid:11)(cid:1) (cid:24) and the negative of the decay rate along
d [asymptotic axis of dilatation] is not a creator of fronts
(cid:11)(cid:1) (cid:24), respectively,
c but a collector of fronts; for though frontogenesis may
occur anywhere in the field, the line of frontogenesis
1
D(cid:1)(cid:3)|(cid:7)(cid:1)(cid:17)d(cid:1)2(cid:8)D(cid:9)(cid:3)E2(cid:2)(cid:8)2(cid:4)1(cid:10)2(cid:10), (cid:3)21a(cid:4) will move toward this axis, and the resulting front will
be found at this axis, or close to it” (Petterssen’s em-
1 phasis).
C(cid:1)(cid:2)(cid:3)|(cid:7)(cid:1)(cid:17)c(cid:1)2(cid:8)(cid:2)D(cid:9)(cid:3)E2(cid:2)(cid:8)2(cid:4)1(cid:10)2(cid:10). (cid:3)21b(cid:4) a. Examples
The asymptotic dilatation rate, D, represents the long- Because the flow in Fig. 1 possessed no vorticity, by
term rate at which the distance between two adjacent (18a)and(18b),asymptoticaxesofdilatationandcon-
parcels increases for steady-state wind fields that vary traction align along the axes of dilatation and contrac-
linearly with the spatial coordinates. Conversely, the tion,respectively.Thus,theairstreamboundaryoccurs
asymptoticcontractionrate,C,representsthelong-term where the axis of dilatation occurs.
rateatwhichthedistancebetweentwoadjacentparcels Next, consider the flow fields shown in Fig. 2, the
decreases,giventhesameconditions.Weusetheterm same as those in Fig. 1 except for the addition of vor-
asymptotic to describe these diagnostics because as- ticity ((cid:14)(cid:1) 0.6E). Whereas parcel pairs oriented along
ymptotic implies long-term behavior given enough theaxesofdilatationandcontraction((cid:15) and(cid:15))inFig.
d c
time. What constitutes “long term” is a length of time 1 would remain oriented along those axes, in Fig. 2
during which the steady-state (or slowly varying) and parcel pairs oriented along the axes of dilatation and
Taylor-expansion approximations made earlier in this contraction would rotate away from those axes and
paper are valid. Analogous to frontogenesis, these di- eventually reach the asymptotic dilatation axis ((cid:24)).
d
agnosticsrepresentlocationswheretheflowisactingto Clearly, the airstream boundaries lie along the asymp-
createairstreamboundariesinstantaneously.Iftheflow totic axes of dilatation in these two flows. Adding a
issteadyandthewindfieldvarieslinearlyinspace,the constantuniformflowtotheseflowfieldswouldelimi-
asymptoticdilatationaxisistheeventuallocationofthe natethecolpointattheoriginwherethewindspeedis
airstream boundary. zero, but the asymptotic dilatation and contraction
Eventhoughweusetheadjectiveasymptotictorep- axes, as well as the axes of dilatation and contraction,
resentthelong-termbehavioroftheair-parcelpairs,it wouldremainunchanged.Thus,inthesetwoflows,al-
isimportanttohighlightthatthesediagnosticsarecom- though the maximum instantaneous dilatation rate oc-
puted kinematically from instantaneous fields. Al- curs along the axes of dilatation ((cid:15)), the airstream
d
though these expressions are exact instantaneously, boundaries occur along the asymptotic axes of dilata-
they are only estimates of the long-term behavior of tion ((cid:24)). The effect of the vorticity ((cid:14)(cid:1) 0.6E) is to
d
time-varying flow, as shown later in the paper for an rotatetheasymptoticaxisofdilatationabout18°coun-
idealized flow. Application of these diagnostics to real terclockwise from the axis of dilatation and to rotate
nonsteady atmospheric flows is the subject of future the asymptotic axis of contraction about 18° clockwise
research. from the axis of contraction.
Theseexpressionsaresimilartothosefortheinstan- Unliketheaxesofdilatationandcontraction,(cid:24) and
d
taneous rates d and c, (10a) and (10b), respectively, (cid:24) need not be orthogonal (e.g., Fig. 2). Indeed, the
c
except for the addition of the vorticity (cid:14)in (21a) and anglebetween(cid:24) and(cid:24) iszerowhenthemagnitudeof
d c
(21b). Therefore, by including the effect of vorticity, the vorticity equals the deformation, as per (18a) and
the expressions for the instantaneous dilatation and (18b)(e.g.,Fig.3,whichisthesameasFig.1exceptthat
contraction rates become expressions for the asymp- (cid:14)(cid:1)E).Inthiscase(Fig.3),theflowisshearedacross
totic dilatation and contraction rates, respectively. a northeast–southwest-oriented shear line. Since (cid:24) (cid:1)
d
1360 MONTHLY WEATHER REVIEW VOLUME133
FIG. 2. As in Fig. 1, except (cid:14) (cid:1) 0.6E, and solid lines indicate the asymptotic dilatation axis ((cid:24)d) and asymptotic contraction
axis((cid:24)).
c
(cid:15) (cid:9) ((cid:16)/4) by (18a) in this flow, (cid:15) is oriented east– Thesecompetingeffectsaredemonstratedintheideal-
d d
west, although the position of the airstream boundary izedshearflowinKeyseretal.(1988,theirFigs.5and
((cid:24))isorientednortheast–southwest,anorientationro- 6),wherethevorticityequalsthedeformationalongthe
d
tated 45° cyclonically from the axis of dilatation. shearline(x(cid:1)0),producingtheaxesofdilatation45°
This result is consistent with that of Keyser et al. anticyclonically to the shear line.
(1988), Bishop (1996), and Schultz (2004) who showed When the magnitude of the vorticity is greater than
that fronts with vorticity do not align along the axis of the deformation, no axes of zero rotation exist (e.g.,
dilatation, but at some angle cyclonically rotated from Fig.4,whichisthesameasFig.1exceptthat(cid:14)(cid:1)1.2E).
the axis of dilatation. In such a case, the deformation In such a situation, when the flow is nondivergent, the
along the front acting to rotate the isentropes anticy- fluidistrapped(e.g.,Fig.4a).Whentheflowpossesses
clonically offsets the vorticity along the front acting to divergence, parcel pairs will spiral around and ap-
rotatetheisentropescyclonically,asdescribedby(18a). proach each other when D (cid:26) 0 (e.g., Fig. 4b) or spiral
FIG.3.AsinFig.1,except(cid:14)(cid:1)E,andsolidlinesindicatetheasymptoticdilatationaxis((cid:24)d).
MAY2005 COHEN AND SCHULTZ 1361
FIG.4.AsinFig.1,except(cid:14)(cid:1)1.2E.
aroundandseparatefromeachotherwhenD(cid:25)0(e.g., [As an aside, Petterssen (1953), Xu (1990, 1992), and
Petterssen 1956, his Fig. 2.7.2F). The dynamics of par- CunninghamandKeyser(2000,2004)showedthat(cid:19)2is
celsinthiscontexthavebeendiscussedpreviously(e.g., proportionaltothesquareoftheageostrophicvorticity
McWilliams1984;Benzietal.1988;Pierrehumbertand (Q in Cunningham and Keyser 2000).]
Yang 1993; Cunningham and Keyser 2000, 2004). AsimilarresultwasdiscussedbyMcWilliams(1984)
foratwo-dimensionalidealizedsimulationofturbulent
b. Development of the fluid-trapping diagnostic:
equivalent-barotropic flow and Pierrehumbert and
Relationship to the Lyapunov exponent
Yang (1993) for a passive tracer being advected on a
The expressions for D in (21a) and C in (21b) repre- two-dimensional, nondivergent, isentropic surface.
sent the long-term rate at which the distance between Benzietal.(1988),Schubertetal.(1999),andCunning-
two adjacent parcels increases or decreases, respec- ham and Keyser (2000, 2004) also derived similar ex-
tively.DependingonthevalueofDandthesignofE2 pressions. As these studies noted, the relative magni-
(cid:2)(cid:14)2,however,Cmaybepurelyreal,purelyimaginary, tudesofdeformationandvorticity(or,equivalently,the
or complex. sign of E2 (cid:2) (cid:14)2) can be used to distinguish between
Iftheflowisnondivergent,theinterpretationcanbe so-called straining or turbulent regions of the fluid
simplified.Inthiscase,D(cid:1)C,implyingthattherateat dominatedbydeformation(E(cid:25)|(cid:14)|)andso-callededdy
which adjacent air parcels approach each other along or neutral regions of the fluid dominated by vorticity
the asymptotic contraction axis is the same rate at (|(cid:14)| (cid:25) E). This analysis can be extended to flows with
which air parcels separate along the asymptotic dilata- divergence, such that the eddy region will expand in
tion axis, as would be expected in nondivergent flow. divergent flow and contract in convergent flow.
(Because D (cid:1) C, we choose C to represent the growth AsshowninappendixB,expressionsforDandCcan
rate of an airstream boundary to maintain consistency alsobederivedfromcalculatingthetwoeigenvaluesof
with C and c from earlier in this paper.) In nondiver- theflow,(cid:27) and(cid:27),wheretheseparationvectorcanbe
1 2
gentflow,CiseitherrealandnonnegativewhenE2(cid:2)(cid:14)2 expressedasalinearcombinationoftwoeigenvectors:
(cid:18)0orispurelyimaginarywhenE2(cid:2)(cid:14)2(cid:26)0.IfCisreal
and nonnegative, then adjacent parcels separate from (cid:2)r(cid:1)Ae(cid:19)1t(cid:9)Be(cid:19)2t. (cid:3)23(cid:4)
each other (e.g., Figs. 2a and 3a). If C is purely imagi- Written in this form, the eigenvectors point along (cid:24)
nary (e.g., Fig. 4a), these parcels are isolated from the d
and (cid:24), the axes where (cid:7)(cid:1) 0. When either A or B is
surrounding flow (i.e., trapped). Therefore, C can be c
zero,theseparationvectorisconstrainedtomovealong
usedtodescribethebehaviorofair-parcelpairsintime
oneeigenvectordirection.Thisisequivalenttorestrict-
and is dependent on the sign of E2 (cid:2) (cid:14)2.
ing (cid:7)to be zero, implying that (cid:27)(cid:1) (cid:6)| . For nondi-
NotethatD2andC2,ifnondivergent,areequaltothe (cid:11)(cid:1)(cid:24)
vergentflow,theeigenvalueassociatedwiththeasymp-
product of (cid:19) and (cid:19) , what we term (cid:19)2, the fluid-
(cid:9) (cid:2) totic dilatation rate D is positive, and the eigenvalue
trapping diagnostic:
associated with the asymptotic contraction rate C is
1 negative. At large t, the term in (23) with the positive
(cid:11)2(cid:1)(cid:11) (cid:11) (cid:1) (cid:3)E2(cid:2)(cid:8)2(cid:4). (cid:3)22(cid:4)
(cid:9) (cid:2) 4 eigenvaluedominates,andthus,onlyoneeigenvalueis
1362 MONTHLY WEATHER REVIEW VOLUME133
FIG.5.EvolutionofDoswellvortexindiffluentbackgroundflowat(a)0,(b)12,and(c)24h.Potentialtemperature(dashedlines
every2K),streamfunction(solidlinesevery3(cid:28)106m2s(cid:2)1),andrelativevorticity(shadedforvaluesgreaterthan2(cid:28)10(cid:2)5s(cid:2)1,in
incrementsof2(cid:28)10(cid:2)5s(cid:2)1).
needed. The single eigenvalue at large t is called the diagnostic that can be used to assess where the flow is
Lyapunov exponent. characterizedbyeddies(e.g.,trappedflow).Finally,in
In studies of chaotic mixing of geophysical fluids this section, these diagnostics were applied to steady,
(e.g.,PierrehumbertandYang1993;referencesinZeng idealized, linear flows. Applying these diagnostics to a
et al. 1993), the Lyapunov exponent is used as a mea- nonsteady, idealized, nonlinear flow is the subject of
sure of the maximum uncertainty of a parcel’s future the next section.
position as a function of the uncertainty in its initial
position. The Lyapunov exponent (cid:27) represents the
maximum growth rate of parcel-pair separations aver- 4. Application to an ideal cyclone
aged over the fluid. Thus, there is some similarity be-
tween the Lyapunov exponent and our diagnostic D, To illustrate the application of these diagnostic ex-
where(cid:27)islongtermandDisinstantaneous,andwhere pressions from sections 2 and 3 to an idealized atmo-
(cid:27)isaveragedovertheentireflowandDislocal.Thus, spheric flow, we choose a synoptic-scale flow pattern
we can say D represents the instantaneous local featuringbothdeformationandvorticity.Wechoosea
Lyapunov exponent, which is dependent on both time nondivergentframeworkinwhichtoillustratethesedi-
and position. agnostic expressions to simplify the interpretation.
The use of the instantaneous local Lyapunov expo- Nevertheless, these results could be generalized to
nent to investigate the stretching of fluid elements is more complex flows using the full divergent formula-
not new. For example, similar expressions have been tion in this paper. Even in such a nondivergent flow,
developed previously by Batchelor (1952), Vuorela contraction between air-parcel pairs occurs and air-
(1953), and Welander (1955). The instantaneous local streamboundariesform.Afterabriefdescriptionofthe
Lyapunovexponentisalsorelatedtotheefficiencyfac- model, two aspects of the flow are examined: the evo-
tor used in studies of mixing, such as Fairlie et al. lution of parcel pairs in section 4a and the airstream
(1999), where the efficiency factor is equivalent to the boundaries in section 4b.
square of the instantaneous local Lyapunov exponent. Weconstructanidealizedsynoptic-scaleflowpattern
Thus, the asymptotic dilatation rate D, or equivalently representing an extratropical cyclone in a diffluent
the instantaneous local Lyapunov exponent, helps to background flow by embedding an idealized Doswell
bridgethegapbetweenthekinematicframeworkposed (1984) vortex [also discussed by Davies-Jones (1985)
bysynopticmeteorologyandthetheoreticalframework and Doswell (1985)] in nondivergent, diffluent flow as
of turbulent mixing. discussed in Schultz et al. (1998). This flow pattern is
Tosummarizethissection,weshowedthatflowswith placed in a time-dependent, nondivergent barotropic
vorticity would possess airstream boundaries differing model on an f plane, adapted from that of Smith et al.
in orientation from the axis of dilatation, if E (cid:18) |(cid:14)|. (1990).Detailsofthebarotropicmodelandexperimen-
Expressions for the asymptotic dilatation and contrac- tal setup can be found in section 4 of Schultz et al.
tion axes were derived and the asymptotic dilatation (1998). Except for weak numerical diffusion, potential
axis was proposed to represent the location of the air- temperatureisaconservedpassivetracerinthismodel.
streamboundarybetterthantheaxisofdilatation.The Theevolutionofthecycloneandpotentialtempera-
asymptotic dilatation rate is also related to the instan- turefieldisshowninFig.5ofthispaperandinFig.13
taneous local Lyapunov exponent and a fluid-trapping ofSchultzetal.(1998).Attheinitialtime,theflowfield