Forced Waves on a Zonally Aligned Jet Stream · Pedlosky (1986) provides a resume of this ﬂow...

1 JANUARY 2004 73S C H W I E R Z E T A L .

q 2004 American Meteorological Society

Forced Waves on a Zonally Aligned Jet Stream

CORNELIA SCHWIERZ, SEBASTIEN DIRREN, AND HUW C. DAVIES

Institute for Atmospheric and Climate Science, ETH, Zurich, Switzerland

(Manuscript received 11 December 2002, in final form 11 August 2003)

ABSTRACT

The potential vorticity (PV) pattern in the vicinity of the jet stream takes the form of a narrow tube of enhancedPV gradient on the in situ isentropic surfaces. It is asserted that this distinctive structure can serve as a waveguideand a seat for trapped Rossby waves and that a neighboring vortexlike anomaly can trigger such waves and/orinteract strongly with the jet. These conjectures are examined theoretically in an idealized setting comprising afinite-scale vortex forcing of a zonally aligned PV discontinuity. The quintessential dynamics of the vortex’sinfluence upon the PV interface are first elucidated in the linear barotropic b-plane limit, and thereafter otheraspects of the jet–vortex interaction are examined in a hemispheric primitive equation setting using a nonlinearnumerical model.

It is shown that for the selected setting the interface can sustain trapped waves, a strong response is favoredby larger-scale forcing, and a quasi-resonant response can prevail for some ambient flow settings, provided thevortex advects zonally at approximately the Doppler-shifted velocity of a trapped Rossby wave. It is also deducedthat (i) a mesoscale perturbing vortex can retain its coherency despite the deforming effect of the ambient flow;(ii) the enhanced PV gradient can indeed serve as an effective waveguide; and (iii) the backreaction of theinterface perturbations upon a weak mesoscale vortex need not be appreciable, and conversely for a strongersynoptic-scale vortex the interaction can lead to significant deformation of both vortex and interface with atendency for a pairing of the vortex with an oppositely signed anomaly on the distorted interface. Commentsare made on the relationship of the results to observed phenomena.

1. Introduction

A striking and ubiquitous feature of atmospheric flowis a meandering jet stream that is located at a break inthe tropopause and almost girdles the globe. A relatedfeature is the distinctive structure of the accompanyingpotential vorticity (PV) distribution. It takes the formof a band of enhanced PV gradient aligned along thejet and located on isentropic surfaces that transect thetropopause break. This band of enhanced PV gradient(=uPV) constitutes the starting point of the presentstudy.

Prototypical examples of both the instantaneous andthe climatological structure of the band are shown inFigs. 1 and 2. These depictions are in line with earlierperspicacious analyses of these features (e.g., Palmenand Newton 1948; Reed and Danielsen 1960), and serveto emphasize the coalignment of the band with the jet,the weak PV gradient of the ambient atmosphere, andthe strength of the band itself. Coalignment conformswith the following quasigeostrophic relationship (castin scaled isentropic coordinates):

Corresponding author address: Cornelia Schwierz, Institute forAtmospheric and Climate Science, ETH Honggerberg, CH-8093 Zu-rich, Switzerland.E-mail: [email protected]

2= PV ; 2¹ U,u u

where U denotes the flow component directed along thejet axis. In effect the relationship indicates that a localmaximum of =uPV also connotes a maximum in theflow. Homogeneity of the ambient tropospheric PV fieldaccords with the mixing of PV by transient eddies (e.g.,McIntyre and Palmer 1984; Sun and Lindzen 1994), andlikewise the enhanced gradient accords with PV front-ogenesis at the tropopause break accompanying the evo-lution of these eddies (Davies and Rossa 1998). Notealso that the amplitudes of the instantaneous and thetime-mean =uPV bands amount, respectively, to ;503 1026 pvu m21 and ;3 3 1026 PVU m21 (PVU: PVunit) and these values correspond to ;O(100) and;O(10) enhancement of their ambient atmospheric orthe purely b-attributable gradient.

Irrespective of a band’s origin its existence, structure,and amplitude carry important dynamical ramifications.First, Rossby waves propagate on the =uPV field, andthe latter field’s localized structure exerts a direct influ-ence upon the dynamical properties and transmissivityof the waves. Second, the strength of the two types ofbands imply that large-scale adiabatic displacement ofair parcels across the band would produce PV anomaliesof ;7 PVU and ;3 PVU, respectively, and in turn suchanomalies would connote significant modifications ofthe in situ (and far field) flow and thermal distributions.

74 VOLUME 61J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. Instantaneous isentropic PV gradient on 1800 UTC 24 Dec 1999 (shaded, units 1026 PVU m21). (a) On the 320-K isentrope, 2-PVU PV contour and wind speed (contours 50, 70, 80, 90 m s21) overlaid. (b) South–north cross section at 308W. Vertical coordinate: u;PV isolines [1, 2 (bold) 3, 4, 5, 6, 8 PVU]; and 50, 70, 80, 90 m s21 isotachs (dashed) overlaid.

FIG. 2. Mean winter (DJF) isentropic PV gradient for the 15-yr period 1979–93 (ERA-15 data). (a) On 320 K with wind speed (contours30, 40, 50 m s21). (b) Cross section at 758W; 30 and 40 m s21 isotachs (dashed) overlaid. Same as Fig. 1, but note the difference in scaleand magnitude.

These inferences suggest that the band’s structure,allied to its temporal persistence and streamwise length,marks it out as an important tropopause-level dynamicalfeature and a possible waveguide in a variety of flowsettings. Two such settings are (i) the transient responseto and interaction with a juxtaposed meso- or synoptic-scale vortexlike anomaly and (ii) the quasi-steady re-sponse to lower-level planetary-scale stationary forcing.

The motivation and rationale for considering the firstsetting arise from the ubiquity and amplitude of such

anomalies. For example Fig. 3a, which shows the PVdistribution on the 320-K surface for the same instantas Fig. 1, pinpoints the presence of meso- and synoptic-scale positive PV anomalies (features A, B, and C) anda negative anomaly (E) that are all in the vicinity of the=uPV band. [Some anomalies (D and F) are removedfrom the band]. In contrast, for the second setting theeffect of the low-level orographic forcing would onlybe evident at higher elevations as an anticyclonic vor-ticity pattern, and there is a hint of such a signal in Fig.


FIG. 3. (a) Instantaneous PV on the 320-K isentrope at 1800 UTC 24 Dec 1999 (shaded, in PVU) with 2-PVU PV contour (bold black)and synoptic and mesoscale vortex anomalies (labels A–F) overlaid; (b) 15-yr winter (DJF) mean of negative vorticity on 500 hPa (shaded,units 1024 s21).

3a over the northern part of the Rockies. A more vividdepiction of this orographic signal is evident in the time-mean vorticity field of Fig. 3b, which shows that theflow over the Rockies, Greenland, and the Himalayasis characterized by negative vorticity values that arealmost unrivalled elsewhere in the extratropics.

The present heuristic study is designed to exploreaspects of the =uPV band’s influence in the foregoingtwo settings. The approach adopted is based upon firstrepresenting the band in the highly idealized form of alatitudinally and vertically aligned interface separatingtwo regions of uniform (but different) potential vortic-ity. Then the response is evaluated when the interfaceis perturbed by an isolated vortex that is either fixed inspace or advects with the ambient flow.

This idealized configuration has a long pedigree inatmospheric studies (Platzman 1949) and has been in-voked in several recent studies (Polvani and Plumb1992; Nakamura and Plumb 1994; Ambaum and Verk-ley 1995; Swanson et al. 1997; Pomroy and Thorpe2000). It has the merit of restricting Eulerian in situadiabatic PV changes to the neighborhood of the inter-face and the vortex and, moreover, in a simple two-dimensional setting it has been shown that the interfacecan sustain trapped Rossby waves. The configurationhas also been adopted in oceanographic studies to modelstrong currents such as the Gulf Stream and such acurrent’s meanders and interaction with an isolated vor-tex (see Bell 1990; Pratt et al. 1991 and references there-in). In particular it has been shown (Bell 1990) that apoint vortex can excite a resonant response of thetrapped wave, and that a backreaction of perturbations

on the interface induces a ‘‘latitudinal’’ displacement ofthe point vortex.

The paper is organized as follows. In section 2 thequintessential dynamics of a finite-scale vortex’s influ-ence upon the interface is examined in a simple linearb-plane setting. This setting is designed to complement,elaborate, and extend the results of earlier studies (inparticular that of Bell 1990) and to concomitantly pro-vide a transparent illustration and interpretation of thelinear ingredients of the vortex’s influence. In section 3the nature of this influence is tested in the more realisticbut still idealized atmospheric setting of nonlinear hemi-spheric primitive equation flow. This setup permits ex-plicit examination of the jet–vortex interaction for vor-tices of different spatial scale, amplitude, and origin.Finally, in section 4 the derived results are placed inthe context of extant studies of jet–vortex interactionand quasi-steady forcing of planetary- and subplanetary-scale atmospheric waves.

2. Interfacial waves on a b plane

a. The flow setting

Quasigeostrophic flow of a homogeneous fluid on ab plane with bottom topography h 5 h(x, y) and a rigidupper lid at a height H (see Fig. 4) satisfies the followingpotential vorticity equation

D fg 0j 1 h 1 by 5 0. (1)5 6Dt H

Here q 5 {j 1 ( f 0/H)h 1 by} is the potential vorticity,


FIG. 4. Schematic diagram to illustrate the analytic model setup,terms, and symbols. (top) A vertical section at y 5 d. See text fordetails.

Dg/Dt is a pseudo-Lagrangian derivative following thegeostrophic flow, j the vorticity, and c denotes the cor-responding geostrophic streamfunction (so that j 5¹2c). Pedlosky (1986) provides a resume of this flowsetting.

A basic state is selected to comprise a zonal flow overflat terrain with (as stated earlier) a zonally aligned in-terface located along y 5 0 separating two semi-infinitedomains of uniform but different values of quasigeo-strophic potential vorticity, 5 6P/2. The correspond-qing zonal flow field is given by

P 12U 5 U 2 sgn y 1 by for y " 0, (2)0 2 2

with sgn 5 sign( ). Small perturbations of this basicqstate satisfy a linearized form of Eq. (1):

] ] ]q1 U q9 1 y9 5 0, (3)1 2 1 2]t ]x ]y

where y9 denotes the perturbation meridional velocity fieldand q9 is the perturbation potential vorticity given by

f0q9 5 j9 1 h. (4)H

b. Free waves

In the absence of topography the foregoing setting isstandard, and we briefly recapitulate previous results.Wave perturbations exist with the streamfunction in thetwo domains taking the form

2(sgn)kyc9 5 Ae sin(kx 2 vt) for y " 0,

with (v, k) denoting, respectively, the frequency and thezonal wavenumber. The dispersion relationship is ob-tained by integrating Eq. (3) across the PV interface,and is given by (see also Bell 1990; Swanson et al.1997)

Pv 5 U k 2 . (5)0 2

Thus, the natural modes of the system are trapped onthe interface and their latitudinal decay scale is deter-mined by the zonal wavenumber k. Consistent with theunderlying Rossby wave character (i.e., propagation ona PV gradient) the individual modes propagate westwardrelative to the basic-state velocity field U0 at the inter-face. However, in contrast to Rossby waves on a uniformflow, the zonal group velocity of these waves is givenby U0 and is independent of k. Also, note that the dis-persion relationship depends exclusively upon the in situvelocity and the vorticity jump at the interface, and notupon the other details of the background basic state.

Note further that the steady-state wave is character-ized by the wavenumber ks such that

Pk 5 . (6)s 2U0

Note that for U0 ; 30 m s21 and mean basic-state vor-ticity values (P/2) in the broad range (3–6) 3 1025 s21,the value of ks equates in spherical geometry to a zonalwavenumber m in the band of m ; 4–8. [A comparablerelationship to Eq. (6) is derived in section 3a for thespherical geometry configuration.]

Thus, the simple two-zone setting can sustain dynam-ically trapped Rossby waves, and moreover their exis-tence and lateral confinement implies that a forcing inthe far field could excite (and possibly resonate with) atrapped wave on the interface. For a stationary forcingfield, resonance would be expected to prevail for awavenumber k satisfying Eq. (6). Alternatively for aforcing field advecting steadily eastward with a velocityU* the corresponding wavelength is given by

Pk* 5 . (7)s 2(U 2 U*)0

In the next section we examine and illustrate the dy-namics of a setup with a stationary vortexlike forcingfield.

c. Isolated topographic vortex forcing

Consider now the setting with the prescribed basicflow state [i.e., satisfying Eq. (2)] perturbed by tra-versing over an isolated topographic feature. In effectthis setup represents the forcing of waves on the inter-face due to a spatially fixed topographically forced vor-tex. The overall dynamics is now governed by Eq. (3),


and the perturbation streamfunction c9 satisfies the re-lationship [see Eq. (4)]

f02¹ c9 5 q9 2 h.H

It follows that the streamfunction is composed of twocomponents (say, ca and cb), representing respectivelya (possibly) time-dependent contribution due to pertur-bations of the interface (the q9 term) and time-indepen-dent topographically induced component (the h term).

Consider in turn the form of these two components.The first component can be written as a Fourier seriesof waves, each of the form

2(sgn)kyc 5 a(t)e cos[kx 1 «(t)] (8)a

with (a, «) denoting the time-dependent amplitude andphase of the kth wavenumber.

For the second (h) term consider an axisymmetricmountain of radius R located at a distance d (d . R)from the PV interface (see Fig. 4). The mountain-in-duced circumferential velocity V (r) at a radius r takesthe form

KV (r) 5 2 for r . R, (9)

r

where K is a measure of the orographically inducedvortex strength such that

1 f0 2K 5 h R01 22 H

for a top-hat-shaped mountain of height h0. The asso-ciated perturbation meridional velocity at the interfacecan be written successively (using of the Fourier sinetransform) as

Kxy9 5 2b 2 2(x 1 d )

2kd5 2K e sinkx dk.EThus the contribution of the kth wavenumber to themeridional velocity at the interface can be written in theform

2kdV9(k) 5 2Ke [sin(kx 1 «) cos«b

2 cos(kx 1 «) sin«]. (10)

Insertion of c9, derived from using Eqs. (8) and (10),into Eq. (3) followed by integration across the interfaceat y 5 0 yields the following coupled set of equationsfor the time evolution of the kth wave component (cf.Davies and Bishop 1994):

]a /]t 5 2Dsin« ,k k

a ]« /]t 5 2ga 2 D cos« (11)k k k k

with

2kdPKe PD 5 , g 5 U k 2 . (12)02k 2

The first equation (11) prescribes the growth of the in-terfacial wave due to the vortex forcing. Growth is fa-vored by a strong PV jump at the interface, a smallwavenumber, proximity of the vortex to the interface,and an appropriate phase alignment (the optimum pre-vails for «k 5 2p/2). The second equation (11) indicatesthat the net phase change arises from the differencebetween the westward propagation of a free mode (theg term) and the vortex-induced change. We note in pass-ing that the nonlinear feedback due to the interfacialwaves acting upon the topographically induced com-ponent is excluded in the linearized system representedby Eq. (3), but we return to consider this aspect insection 3.

It is instructive to consider two limit forms of thecoupled set.

(i) A time-independent response:In this case each wavenumber is prescribed by

« 5 0 or p and a 5 |D /g | .k k

Thus, a steady state prevails when the westward prop-agation of the Rossby wave trapped on the interface ismatched by an opposing tendency attributable to thevortex forcing (see also Bell 1990).

(ii) The resonant response:This response prevails for a wavenumber satisfying

the relationships

« 5 2p/2, g 5 0,k

so that

a (t) 5 D · t.k

These criteria equate to the optimum configuration ofthe vortex and wave for the latter’s growth (i.e., «k 52p/2), and it is sustained because the correspondingfree mode is stationary relative to the forcing (g 5 0,cos«k 5 0).

A more general solution of the coupled set [Eqs. (11)]can be found on noting that the set possesses the fol-lowing integral invariant:

2a Dk 1 a cos« . (13)k k2 g

Hence the evolution for an initially unperturbed inter-face, ak(0) 5 0, is given by

1 1 ga (t) 5 sin(mD · t) for m 5 (14)k m 2 D

with the accompanying phase evolution given by

cos« 5 2ma .k k (15)


FIG. 5. (a) Initial velocity (dotted) and perturbation velocities after 2 (light) and 5 days (bold); units along x axis are 1000 km. (b)y9aAmplitude evolution for different modes k in the domain of length Lf 5 2pR cosf, with R (earth’s radius) and Dk 5 2p/Lf: 3Dk (dotted),Dk (bold) and 5Dk (light). Time in days. (c) Energy evolution [normalized, E (t)/E (0)] for the two extreme cases kr 5 3Dk (bold) and kr 53.5Dk (light). Also shown (dotted) is a function proportional to t2 (for t $ 10 days). (d) Spatial flow pattern after 5 days.

It follows from Eqs. (8), (14), and (15) that the ca

component of the streamfunction can be rewritten as

12(sgn)ky(c ) 5 2 e {cos(kx) 1 cos[kx 1 2«(t)]}.a k 1 22m

(16)

This neat two-term formulation of the waveguide’s con-tribution was pointed out to us by a reviewer. The firstterm is stationary and is a direct modification of thevortex’s contribution to the streamfunction, and the sec-ond is a propagating wave whose instantaneous velocityis given by (22«/k).

Illustrations of the flow evolution for an elliptic-shaped mountain centered 1000 km poleward of theinterface and with a major axis of 1000 km are provided

in Fig. 5. (Note also that H 5 10 km, P 5 5 3 1025

s21, h0 5 2 km, and a b plane with an east–west spanof 27 000 km.) Figures 5a–c show the temporal devel-opment of the flow, and Fig. 5d the spatial pattern after5 days. These depictions show a train of interfacialwaves that evolve downstream with the ambient flow(sic. the group velocity), and their wavelength (;8400km) matches that of the stationary wave [Eq. (6)]. Thewave train is well developed after 5 days, and after some10 days the wave train approaches the ‘‘longitude’’ ofthe topography from the west.

Thereafter there is evidence of constructive inter-ference. Clearly the nature of the interference—con-structive or destructive—depends crucially on whetherthe b plane’s east–west span equates to an integer mul-tiple of the stationary wavelength. This behavior is


illustrated (see Fig. 5c) by examining the time trace ofnormalized wave energy on the interface, E (t)/E (0),where

2E(t)| 5 {[u9(x, t) 1 u9(x, t)]y50 E a b

21 [y9(x, t) 1 y9(x, t)] } dxa b

for parameter settings (P, U0) corresponding to a po-tentially resonant (g 5 0) setting and a nonresonantsetting. Both experiments show a similar short-time evo-lution, but the longer-time development is markedly dif-ferent. The energy for the ‘‘resonant’’ mode grows liket2 in the longer-time limit, whereas the amplitude of theoff-resonant mode merely oscillates with time.

3. Interfacial waves on a hemisphere

The analysis of the previous section does not accountfor effects beyond b-plane geometry, quasigeostrophy,linearity, and a discontinuous PV interface. In this sec-tion we consider first the form of free waves on a baro-tropic PV discontinuity in spherical geometry and thenexamine the robustness of the dynamics using a non-linear primitive equation model with a finite-width PVtransition zone.

a. Free waves

Consider the analogue of the flow setting of section2b but now in a hemispheric geometry. The basic-stateabsolute vorticity is set to a value of P poleward andzero equatorward of a specified latitude f0. The cor-responding zonal velocity U expressed in terms of U 5U cosf is then given by

2Pa(1 2 sinf) 2 Va(1 2 sin f) for f . f0U 525Pa(1 2 sinf ) 2 Va(1 2 sin f) for f , f0 0

with a and V denoting the radius and angular velocityof the earth. Assuming a zero-perturbation meridionalvelocity at the equator, then the streamfunction in thetwo domains is given by

2mpAe sin(ml 2 vt)c 5

1mp 2mp5AG(e 2 e ) sin(ml 2 vt).

Here (l, m) refer, respectively, to longitude and thezonal wavenumber, p 5 tanh21 (sinf), and the amplitudemodifying factor G is such that G 5 2e /(2 sinhmp0).2mp0

The dispersion relationship takes the form

Pv 5 mV 2 , (17)0 1 1 cothmp0

where V0 is the angular velocity of the zonal flow atthe latitude of the PV interface.

The correspondence with the earlier b-plane disper-sion relationship is brought out on noting that for aninterface located in midlatitudes

cothmp . 10

provided m $ 3. In this limit the waves are in effectlatitudinally trapped to the interface, the dispersion re-lationship reduces to

Pv ø mV 2 , (18)0 2

and the azimuthal group velocity ]v/]m is approximatedby V0; that is, it is determined by the angular velocityat the interface.

In the same limit the equivalent for spherical ge-ometry (with m $ 3) of the resonant waves prescribedby Eqs. (6) and (7) are given by

m 5 P /V , (19)s 0

m* 5 P(V 2 V*). (20)s 0 0

Here Eqs. (19), (20) apply, respectively, to a stationarysetting and a forcing field that advects eastward with anangular velocity . Thus, in spherical geometry res-V*0onance can prevail if the flow setting is such that theratio of P and either V0 or (V0 2 ) deliver an integralV*0number corresponding to the zonal wavenumber m. Inthis context note that P and V0 are not interdependentsince

(P/V ) 5 [1 1 (V/V )] (1 1 sinf )0 0 0

or equivalently21 21(P/V ) 5 [(1 1 sinf ) 2 (V/P)].0 0

b. Primitive equation simulations

Idealized numerical simulations are performed in aprimitive equation (PE) framework to examine jet–vor-tex interaction on the hemisphere. The basic-state set-ting resembles that used in section 2.

1) MODEL SETUP

Simulations are undertaken with an adiabatic versionof the so-called Europa Model (EM) of the GermanWeather Service (Majewski 1991). This gridpoint modelis used herein in a pole-centered, rotated-grid configu-ration on a quasi-hemispheric domain and operated witha 18 (;100 km) resolution, 27 vertical levels, a free-slip lower boundary condition, a rigid upper-lid, weakfourth-order horizontal diffusion, and a relaxation zonein the equatorial region.

2) BASIC STATE

One component of the basic state in the extratropicsis a barotropic zonal flow associated with two regionsof uniform PV separated by a zonally aligned transitionzone that extends over some 48 of latitude in the formof a column of strong horizontal PV gradient. The struc-


FIG. 6. (a) Zonal velocity profile (solid line, units m s21) and vorticity discontinuity [dashed, values 7.25 and 15 (31025 s21)] of thebarotropic initial state for the PE model operated on the sphere. The PV discontinuity is located at 47.58N. (b) Initial positive vorticityanomaly (shaded, units 1024 s21) located north of the PV discontinuity (solid line).

ture of the resulting barotropic jet and the accompanyingvorticity distribution are shown in Fig. 6. This initialconfiguration bears comparison with the atmosphericsettings discussed earlier that were portrayed in Figs.1–4.

The other component comprises a vortexlike structurethat takes one of the following three observationallymotivated forms: (i) a single interior positive vorticityanomaly of meso-a scale (radius a0 ; 300 km) em-bedded within the high PV region (Fig. 6b) that is freeto advect with the ambient flow and located at tropo-pause elevations at various distances from the transitionzone, (ii) a counterpart positive or negative anomaly ofsynoptic-scale (a0 ; 700 km) (Fig. 9), and finally (iii)a hybrid setup with both an orographically bound neg-ative anomaly and an advecting surface-based positivethermal anomaly.

These three configurations relate to different physicalsettings that were alluded to in the introduction (see alsoFig. 3). Consider first the setting in case I with a positivevortex located poleward of and adjacent to a jet stream(cf. features marked A, B, and C in Fig. 3a). Both theseingredients have been implicated in the surface cyclo-genesis. Incipient wavelike meanders of the jet stream(sic. =uPV waveguide) are often taken to be indicativeof the upper-level signature of a troposphere-spanninggrowing baroclinic wave system (sic. baroclinic insta-bility), and likewise a tropopause-level positive PVanomaly is noted as a possible precursor of surface fron-tal wave development. In effect these two categoriesbear comparison, respectively, with so-called Types Aand B events of cyclogenesis.

In the present setup, consideration of interlevel in-teraction is eschewed and the focus was on single-levelinteraction. This is motivated by the recognition thatfrequent close proximity of a tropopause-level positivePV anomaly with the jet stream suggests that their in-teraction can indeed be a significant if not a dominant

feature of the short-term development (see, e.g., Fehl-mann and Davies 1999).

The setting in case II with a negative synoptic-scalevortex anomaly located within the high PV domain cor-responds to the not-infrequent atmospheric setting whena large-scale upper-tropospheric air mass associatedwith a strong high pressure system has been sequesteredinto the stratosphere (cf. features marked D and E inFig. 3a). The reverse can also occur with sequestrationof stratospheric air into the upper troposphere in theform of an elongated PV streamer that can break upinto a compact vortexlike structure (Appenzeller andDavies 1992). This is illustrated by the feature markedF in Fig. 3a.

Included in case III is the ingredient of perturbationson the =uPV band induced by a quasi-steady anticy-clonic vorticity above topography (cf. the negative vor-ticity over the Rockies and Greenland in Fig. 3b). Otherpossible sources of such a steady forcing field wouldbe regions of anomalous diabatic heating associatedwith SST anomalies in the subtropics. Again for a strongresponse the amplitude and scale of the forcing need tobe appreciable and sustained, and concomitantly the=uPV band needs to be coherent. More trenchantly, onthe longer time scale the possibility of a resonant re-sponse cannot be excluded a priori.

3) INTEGRAL DIAGNOSTIC

For diagnostic purposes it is useful to note that in-viscid barotropic flow on the hemisphere possesses thefollowing integral invariant:

(j 1 f ) sinf dS 5 constant. (21)EES

This invariant has direct ramifications for our particularsetting of two zones of uniform PV and a single un-


FIG. 7. Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 1024 s21) for a moving vorticity anomaly actingon a vorticity gradient at 47.58N. Latitude circles shown for 408 and 808N.

constrained vortex. It follows that growth of wave per-turbations beyond the transition zone would need to becountered by a compensating latitudinal movement ofthe vortex (cf. Bell’s backreaction). Thus, for examplea large amplitude wave pattern on the interface wouldbe accompanied by the movement of a positive (neg-ative) vortex toward (away) from the interface.

A description and discussion is now provided of thesimulated patterns.

4) CASE I: MESOSCALE VORTEX FORCING

An interior, meso-a scale, positive PV anomaly islocated some 148 poleward of a PV interface at 47.58N.For this setting the component of the meridional velocitywithin the transition zone attributable to the vortex forc-ing is comparatively weak and the response should beat most quasilinear.

Figure 7 shows the perturbation vorticity field on the


500-hPa surface for the period 60 to 300 hours. In ac-cord with linear theory a trapped wave train of PVanomalies develops within the transition zone of highPV gradient. It evolves downstream with the back-ground in situ flow speed, and its wavenumber (m 56) conforms to that expected from Eq. (20).

Consider the amplitude and structure of the inducedpatterns. First, note that the wave train is confined tothe near neighborhood of the PV transition zone. Ineffect, nonlinear effects are small and the perturbationenergy is exported downstream rather than being avail-able ab initio to build up the in situ perturbation am-plitude. Indeed the wave train’s vorticity anomalies are;0.25 3 1024 s21 and this amounts to only a third ofthe vorticity difference across the transition zone.

Second, note that the backreaction (Bell 1990) ofthe wave train upon the vortex’s latitudinal movementappears to be weak. Indeed the vortex advects east-ward at a quasi-constant latitude in the time periodbeyond 60 hours. From a synoptic-dynamic stand-point it is evident that the prevailing configuration issuch that the isolated vortex is located only slightlyto the west of the first (negative) vorticity perturbationon the wave train, and concomitantly the meridionalvelocity at the vortex center is comparatively weak(not shown). In effect, the realized phase is not sup-portive of a strong backreaction of the wave trainupon the vortex’s movement. From a global-integra-tive standpoint further insight on the weak backreac-tion can be gleaned from Eq. (21). The wave train’sweak amplitude and its confinement to the vicinity ofthe transition zone implies that the lateral movementof the vortex required by Eq. (21) does not need tobe appreciable.

Third, the isolated vortex retains a striking coherencethroughout the simulation despite the latitudinal shearof the ambient flow. This is another demonstration thatnonlinear dynamical effects can enable a vortex to coun-ter the deforming effects of an ambient shear (Meachamet al. 1990).

Fourth, the wave train encircles the globe after 285h and this is followed by a measure of constructiveinterference with the m 5 6 pattern persisting at leastuntil 480 h (not shown). This longer-term behavior isthe manifestation of the resonant buildup of the wavetrain’s amplitude and accords with the linear theory ofthe previous section. However, it also points to the the-ory’s limited validity and applicability. A temporalbuildup of the wave amplitude also requires [see Eq.(21)] a corresponding movement of the vortex awayfrom the critical latitude [defined by Eq. (20)], and thisimplies that the longer-term response can at most be anonsingular resonance.

The sensitivity of the nature of the response to thevortex’s latitude [cf. Eq. (20)] is brought out by con-ducting three simulations with the vortex placed at dif-ferent initial latitudes corresponding approximately toconstructive interference at zonal wavenumbers of m 5

5 and 6, and an intermediate destructive setting. Figure8 shows the perturbation vorticity pattern on the 500-hPa surface for each simulation at a time instance (t 5240 h) shortly before the wave train has encircled theglobe and at a later instant (t 5 315 h). It is evidentthat destructive interference prevails downstream in theoff-resonant setting.

5) CASE II: SYNOPTIC-SCALE VORTEX FORCING

Counterpart simulations to case I are performed withfirst a positive and then a negative synoptic-scale vortex(a0 ; 700 km). The scale and strength of the anomaliesbetoken significantly larger initial velocity field at theinterface that is attributable to the vortex, and therebythe likelihood of nonlinear effects influencing the sub-sequent flow evolution. Figures 9a and 9b show, for thepositive and negative anomalies respectively, the per-turbation PV fields at tropopause elevations for the ini-tial time and at 24, 36, 48, and 72 h.

The flow evolution differs markedly for the two cases,but there are three common underlying dynamical ef-fects. First, both vortices undergo a strong asymmetricdeformation under the influence of the ambient flow’slateral shear while perturbations evolve on the interface.During this phase the more equatorward portion of thevortex becomes distended longitudinally to form anelongated filament.

Second, the distorted vortex’s own velocity signal in-duces in the case of the positive (negative) anomaly apoleward (equatorward) displacement of the filamentaway from (toward) the PV interface and thereby servesto weaken (enhance) the filament’s interaction withanomalies evolving on the interface. The sign of thedisplacement is in accord with Eq. (21).

Third, there is a tendency for the vortex to becomealigned with an adjacent oppositely signed anomaly onthe interface whereas, in contrast, the filament becomesjuxtaposed to or coalesces with a similarly signed in-terface anomaly.

In the subsequent evolution (not shown) the positivevortex sheds its filament, regains its circular shape at amore poleward latitude, and thereafter a wave train de-velops on the interface as in case I. In contrast thenegative anomaly approaches the interface and even-tually is smeared out. Thus the finite amplitude devel-opments in the two cases are radically different but con-sistent with the integral invariant [Eq. (21)].

6) CASE III: HYBRID VORTEX FORCING

In this third case Greenland-scale and height topog-raphy is located poleward of the interface. The initialconditions are such that the perturbing effect of the to-pography induces two vortexlike structures that subse-quently proceed to influence the PV transition zone. Onevortex is the topographically bound anticyclone (cf. theforcing discussed for the b-plane setting of section 2).


FIG. 8. Perturbation vorticity response at wavenumbers (top) m 5 5, (bottom) m 5 6, and (middle) for a nonresonant case for the integrationtimes (left) 240 h and (right) 315 h. The different wavenumbers result from the particular location of the anomaly (bold contours 0.1, 0.3,0.5 PVU) relative to the waveguide (65.58, 638, and 61.58N, respectively). Cylindrical projection of the region 408–758N, 1808W–1808E.

In effect this vortex results from the fission of the incidentpotential vorticity into a negative relative vorticity com-pensated by enhanced thermal stratification as the flowsurmounts the terrain (cf. Schwierz and Davies 2003).

The second vortex is shed off the topography in thefirst phase of the simulation and advects with the am-bient flow. It is a surface-based positive thermal anom-aly resulting from the imposed initial conditions of uni-form stratification and possesses a concomitant positivevorticity.

In this case (Fig. 10) a composite pattern developswith a short-wavelength wave train (m ; 6) evolvingdownstream of the advecting vortex and a longer-wave-length and weaker-amplitude wave train (m ; 3) evolv-ing downstream from the stationary topographicallybound anticyclone. The difference in wavelength is inaccord with Eqs. (19) and (20) with the smaller relativevelocity of the advecting vortex associated with a short-

er-wavelength wave train. The difference in vortical am-plitude accords with the reduction of the energy fluxtransferred to the stationary wave train.

An alternative depiction of the development isshown in Fig. 11 in the form of a Hovmoller diagram.The development occurs in three phases. An initialdevelopment of the longer-wavelength features fol-lowed by the codevelopment of the almost spatiallyseparate wave trains, and finally the emergence of aninterference pattern (with m ; 5) after the short-wave-length wave train approaches the longitude of the to-pography.

7) FURTHER REMARKS

In case I the simulations with the mesoscale vortexindicates that a major distortion of the jet conducive todeep cyclogenesis requires stronger vortex forcing. Such


FIG. 9. (a) Positive and (b) negative synoptic-scale vortex anomalies (a0 ; 700 km, lat0 ;618N) acting on a discontinuity at 47.58N. Perturbation isentropic PV on 320 K for 0, 24, 36,48, and 72 h. Contour interval is 0.15 PVU. Vortex anomaly amplitudes at initial time are 12and 21.5 PVU, respectively.


FIG. 10. Time evolution (60–300 h) of the perturbation vorticity response (shaded, units 1024 s21) for both an orographic stationaryvorticity source (dashed contours) and a moving vorticity anomaly (solid contours) acting on a vorticity gradient at 47.58N.

forcing with concomitant in situ development, ratherthan energy propagation downstream along the wave-guide, could ensue [see Eq. (21)] with a pairing of thepositive anomaly and a negative anomaly created on the

interface and their subsequent poleward displacement(cf. case II).

In case II the simulations involved strong nonlineareffects and there was a tendency for the negative vortex


FIG. 11. Vorticity Hovmoller diagram at 47.58N for a 480-h PEsimulation (PV discontinuity located at 47.58N, orography centeredat 458W).

to move toward, and conceivably for a larger amplitudevortex to be extruded into the low PV domain. Such anexpulsion is often observed on isentropic PV charts andthe result has a bearing upon the assessment of netstratosphere–troposphere exchange of mass and chem-ical constituents.

For case III the simulation showed that the =uPVband can indeed serve as a planetary-scale waveguidetransmitting the effects of the forcing to the far fieldalong a well-defined and confined path. Note on theother hand (cf. case II) that a large amplitude forcingcan deform the band irreversibly and thereby hinderpropagation downstream.

4. Final remarks

The present study was predicated upon the assertionthat the distinctive structure of the PV pattern in theneighborhood of the tropopause-level jet stream can ex-ert a major influence upon large-scale atmospheric flow.In particular, it was conjectured that the accompanyingband of highly enhanced =uPV can serve as an effectivewaveguide for large-scale atmospheric flow and the seatfor trapped Rossby waves forced by juxtaposed isolatedPV anomalies and/or larger-scale orography.

These conjectures were tested in an idealized settingcomprising a vortexlike forcing of a zonally aligned PVdiscontinuity. First, the quintessential dynamics of thevortex’s influence upon the PV interface were elucidatedin the linear barotropic b-plane limit. In this settingtrapped Rossby waves can be sustained on the interface,and our analysis pointed to the factors required to pro-duce a large response to the forcing. These includedplanetary- or subplanetary-scale steady-state forcing orsynoptic-scale forcing for an advecting vortex, the pos-

sible establishment of a resonant response in the longer-time frame for a suitable large-scale flow setting and avortex moving zonally at the Doppler-shifted velocityof a trapped Rossby wave.

Thereafter further aspects of the interaction were ex-amined in a hemispheric primitive equation setting usinga nonlinear numerical model. Simulations performedwith various forcing configurations served to lend cre-dence to nonsingular resonant interaction in a settingwith weak forcing and to indicate the disparate rangeof development patterns in a setting with stronger forc-ing.

Clearly the generality and applicability of the derivedresults is limited by the selection of a simple idealizedsetting. In reality, the domain of enhanced PV gradienton isentropic surfaces is a narrow tube that meandersaround the globe and is often breached by large-am-plitude breaking waves rather than taking the form ofa zonally and vertically aligned interface (see Figs. 1and 3). Moreover, the presence of both the tropopause-level tube of enhance PV gradient plus the narrow zoneof enhanced surface baroclinicity (excluded herein) con-stitute two waveguides and, as in classical baroclinicinstability, perturbations on these waveguides can in-teract to their mutual enhancement.

Nevertheless the derived results do bear comparisonwith and shed light on observed phenomena. To set theresults in context it is appropriate to note that the presentstudy’s spatially confined Rossby waves are trapped in-ternally by the atmosphere’s dynamical (PV) structureand therefore differ intrinsically from classical Rossbywaves propagating on comparatively smooth PV fields.Moreover, the markedly weak isentropic PV gradientsaway from the jet limits the efficacy of meridional wavepropagation and the concomitant energy flux of con-ventional Rossby waves.

For synoptic-scale flow one inference is that the ini-tiation of major upper-level wave development wouldbe favored by a large amplitude, suitably scaled, andlatitudinally located PV anomaly. The latitude corre-sponds to a location where the vortex would advecteastward with a zonal velocity matching that of thetrapped wave. It is pertinent to record that such anom-alies are more likely to be located at the tropopauselevel. Again, subplanetary-scale (m ; 3–7) steady-stateforcing could be effective for a suitable large-scale flowsetting. In this case an informative and illustrative de-piction of the markedly different nature of wave prop-agation in superrotational flow versus one with a lo-calized jet structure is shown in Branstator (1985a, Fig.3). This result points to the efficacy of the =uPV band’srole as a waveguide for subplanetary–scale waves. Thepropagation on a PV waveguide is also in accord withsimulations performed with a barotropic zonally varyingbasic state derived from the climatological NorthernHemisphere 300-hPa field (Blackmon et al. 1983; Bran-stator 1985b, 2002) and with the more regularly struc-tured waves observed on the Southern Hemisphere jet


(Simmons et al. 1983). In a related vein it is pertinentto note that the subplanetary scale carries most of thesignal of the atmosphere’s interannual variability, andMassacand and Davies (2001) interpreted the observedpatterns of this variability directly in terms of wavepropagation along the time-mean =uPV band—in effectthe PV waveguide. Hence, the analysis outlined herelends further support to a complementary interpretationof the establishment of teleconnection patterns.

Acknowledgments. The authors would like to expresstheir gratitude to Rene Fehlmann for providing the PVinversion routine and support with an early version ofthe simulation setup and to Daniel Luthi for expert helpwith the EM model. The neat deduction leading to Eq.(16) was pointed out to us by one of the reviewers. Wethank ECMWF and MeteoSwiss for providing access tothe meteorological data. This study was conducted inpart with funding from the NCCR Climate program ofthe Swiss National Science Foundation.

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