Tropical Cyclogenesis via Convectively Forced Vortex ...

3176 VOLUME 55J 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

q 1998 American Meteorological Society

Tropical Cyclogenesis via Convectively Forced Vortex Rossby Waves in aThree-Dimensional Quasigeostrophic Model

MICHAEL T. MONTGOMERY AND JANICE ENAGONIO

Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

(Manuscript received 12 May 1997, in final form 28 January 1998)

ABSTRACT

This work investigates the problem of tropical cyclogenesis in three dimensions. In particular, the authorsexamine the interaction of small-scale convective disturbances with a larger-scale vortex circulation in a nonlinearquasigeostrophic balance model. Convective forcing is parameterized by its estimated net effect on the potentialvorticity (PV) field. Idealized numerical experiments show that vortex intensification proceeds by ingestion oflike-sign potential vorticity anomalies into the parent vortex and expulsion of opposite-sign potential vorticityanomalies during the axisymmetrization process. For the finite-amplitude forcing considered here, the weaklynonlinear vortex Rossby wave mean-flow predictions for the magnitude and location of the spinup are in goodagreement with the model results. Vortex development is analyzed using Lagrangian trajectories, Eliassen–Palmflux vectors, and the Lorenz energy cycle.

Using numerical estimates of the magnitude of PV injection based on previous observational and theoreticalwork, the authors obtain spinup to a 15 m s21 cyclone on realistic timescales. Simulation of a midlevel vortexwith peripheral convection shows that axisymmetrization results in the spinup of a surface cyclone. The axi-symmetrization mechanism demonstrates the development of a warm-core vortex. The relative contribution fromeddy-heat and eddy-momentum fluxes to the warm core structure of the cyclone is investigated.

The vortex spinup obtained shows greater than linear dependence on the forcing amplitude, indicating theexistence of a nonlinear feedback mechanism associated with the vortex Rossby waves.

Building on recent work by several authors, this work further clarifies the significance of the axisymmetrizationprocess for the problem of tropical cyclogenesis. The theory is shown to be consistent with published observationsof tropical cyclogenesis. Further observational and modeling tests of the theory, specific to the dynamics examinedhere, are proposed.

1. Introduction

Tropical cyclones form in the presence of convectivedisturbances in the Tropics. Many such disturbances arepresent in an ocean basin at any given time. However,only a small fraction (about 80 per year) evolve intotropical storms, which may in turn become hurricanes.It is generally accepted that favorable climatologicalconditions for tropical cyclogenesis include the presenceof low-level convergence, the presence of low-level cy-clonic relative vorticity, the absence of strong verticalshear of the horizontal winds, an atmosphere conduciveto deep moist convection, a significant value of the plan-etary vorticity, and sea surface temperatures greater thanapproximately 268C with a deep oceanic mixed layer.About 80% of all tropical cyclones originate in or nearthe monsoon troughs or the intertropical convergencezone; most others form from disturbances in the easterlytrade winds (Gray 1968; Frank 1987).

Corresponding author address: Dr. Michael T. Montgomery, De-partment of Atmospheric Science, Colorado State University, FortCollins, CO 80523-1371.E-mail: [email protected]

The cyclogenesis process examined here builds onthe studies of vortex axisymmetrization and vortexRossby wave dynamics by MacDonald (1968), Guinnand Schubert (1993), Smith and Montgomery (1995),Kallenbach and Montgomery (1995), and Montgomeryand Kallenbach (1997). Guinn and Schubert made anextensive study of the connection between PV or vor-tex Rossby waves and mature hurricane spiral bands,including a simulation of the potential vorticity dynam-ics of convective asymmetries. Montgomery and Kal-lenbach further clarified the structure of radially andazimuthally propagating Rossby waves on a circularvortex, and also examined vortex Rossby wave–mean-flow interactions in a quasi-linear nondivergent model.They hypothesized that initial vorticity asymmetriessuch as those that arise from moist convective forcingwould accelerate the mean tangential winds and pro-posed this interaction as a mechanism for tropical cy-clogenesis. These ideas are analogous to the problem ofthe intensification and maintenance of a large-scale zon-al jet by forced planetary Rossby waves (Shepherd1987), except the pertinent vorticity gradient in our caseis the radial vorticity gradient of the vortex. This papertests the Montgomery–Kallenbach vortex spinup mech-

15 OCTOBER 1998 3177M O N T G O M E R Y A N D E N A G O N I O

anism in a fully nonlinear, three-dimensional quasi-geostrophic balance model. The three-dimensional char-acter of the model allows the investigation of the vor-tex’s eddy-forced secondary circulation, crucial to un-derstanding the origin of the vortex’s warm core.

We examine the problem of vortex development as-sociated with the production of cyclonic vertical vor-ticity by cumulus convection. Such convection could becaused by environmental forcing (Challa and Pfeffer1980, 1990; Pfeffer and Challa 1981; Montgomery andFarrell 1993; Challa et al. 1998) or by mesoscale pro-cesses. Although it has long been known that energyextracted from the underlying ocean and realized as la-tent heat during condensation provides the principal en-ergy source for tropical cyclones, and this process hasbeen simulated by numerical models such as that ofOoyama (1969) and Kurihara and Tuleya (1981), thiswork further elucidates the nonaxisymmetric advectivedynamics that promotes upscale transfer of convective-scale energy to vortex-scale energy. Organization ofconvection by the storm is not necessary in our model,although organization into a tropical cloud cluster ormesoscale convective system is assumed. The cyclo-genesis mechanism proposed here does not require acooperative interaction between convection and thelarge-scale vortex. Cooperative interaction (Smith 1997;Stevens et al. 1997; Ooyama 1982) could enhance thedevelopment process presented here at later stages inthe cyclone life cycle and may become the dominantmechanism by the hurricane stage. For simplicity, wealso ignore the possible effects of the ambient verticalshear.

Our work describes cyclogenesis from a preexistingmesoscale vortex in the presence of convection, such asthat which would be present in a tropical cloud cluster.The initiating vortex in our simulations has a 200-kmradius of maximum tangential winds (RMW) and aninitial maximum tangential wind speed of 5 m s21. Thephysical setup is consistent with typical values for me-soscale convectively generated vortices (Johnston 1981;Bartels and Maddox 1991). Cyclonic vortices of thistype could also be found within the closed flow patternsoccurring in easterly waves (Reed et al. 1977). PVanomalies induced by cumulus convection on the pe-riphery of the vortex assimilate into the parent vortexvia axisymmetrization. Although this process can bedescribed phenomenologically as vortex merger andstripping, we show that it is also useful to characterizeit in terms of the interaction between vortex Rossbywaves and the mean flow. The wave–mean-flow ap-proach, which has proven successful in describing otheratmospheric phenomena such as the quasi-biennial os-cillation and sudden stratospheric warmings (Holton1992), is shown here to yield valid quantitative predic-tions for the magnitude and location of the vortex spinupat the finite amplitudes considered relevant to the cy-clogenesis problem. The formation of a 5-K warm corewithin a reasonable timescale provides compelling ev-

idence that our proposed mechanism may capture theessence of the cyclogenesis process. In addition, we findthat symmetric outbreaks of penetrative convection nearthe center of the preexisting vortex will give strongerand more rapid spinup.

Sensitivity studies of the axisymmetrization mecha-nism reveal the presence of nonlinear feedback in thespinup process, with the increase in tangential velocitygenerally having a greater than linear proportionality tothe forcing amplitude.

The cyclogenesis mechanism proposed here may becontrasted with cyclogenesis due to the interaction andmerger of mesoscale convective vortices (MCVs) pro-duced in the stratiform region of mesoscale convectivesystems in a favorable large-scale environment (Simp-son et al. 1997; Ritchie and Holland 1997; Ritchie1995). Our mechanism focuses on the process by whichPV (or vorticity) anomalies due to moist penetrativeconvection relax to axisymmetry in the presence of apreexisting vortex, such as an MCV or a low-level cy-clonic vorticity anomaly. The difference between thesetwo approaches may lie in the emphasis placed on theimportance of convective versus stratiform heating. Thetwo ideas are not mutually exclusive; in fact, Ritchieand Holland (1997; section 4b) describe an interactionof two low-level circulations in the early stages of cy-clogenesis that may well be an example of our process.In addition, our work demonstrates the usefulness of theunderlying vortex Rossby wave dynamics in describingthe redistribution of convectively induced PV.

In section 2 we summarize the models used in ourstudies. Section 3 describes the basic states and initial-izations used. In section 4 we examine three-dimen-sional vortex axisymmetrization in the presence of con-vection. Section 5 examines vortex spinup in the pres-ence of ongoing convection and describes sensitivitytests of our results. Section 6 reviews the most relevantobservations of tropical cyclogenesis to our theory. Sec-tion 7 summarizes the results of our work and proposesfurther observational and modeling tests of the theory.

2. The numerical models

a. The quasigeostrophic and semispectral models

The chief model employed is a three-dimensionalBoussinesq quasigeostrophic model on an f plane. Heref is the constant Coriolis parameter and, for simplicity,the static stability N 2 is assumed uniform throughoutthe troposphere. Henceforth Q denotes the total quasi-geostrophic PV, u the flow potential temperature (totalminus resting state), z the relative vertical vorticity, andf the flow geopotential. Here, u0 is the surface potentialtemperature, H the depth of the model troposphere, andL the Rossby radius NH/f 5 1000 km. For convenience,all numerical calculations are performed in nondimen-sional units. Denoting nondimensional variables withtildes we let


TABLE 1. Multiplication factors to be used in convertingparameters from nondimensional to dimensional units.

Parameter Factor

Horizontal distance (x, y, r)Vertical distance (z)Time (t)Time period (T)Horizontal velocity (u, y)

106 m104 m104 s

teddy 5 2.91 days100 m s21

Vertical velocity (w)Temperature (u)Geopotential (f)Relative vertical vorticity (z)Potential vorticity (Q)

1 m s21

30 K104 J kg21

1024 s21

0.3 3 1026 m2 s21 K kg21

21(x, y, z, t) 5 (xL, yL, zH, t f ), (1)

H(y , y , w) 5 NH y , y , w , (2)x y x y1 2L

2N Hu02 2 ˜(f, u, Q) 5 fN H , u , Q f . (3)1 2g

Here (x, y, z) are the Cartesian coordinates with z thepseudoheight (Hoskins and Bretherton 1972), t the time,and (y x, y y, w) the zonal geostrophic, meridional geo-strophic, and vertical velocities, respectively. We alsodefine (r, l) to be the cylindrical coordinates with r 50 at the center of the initial azimuthal mean vortex onthe lowest level and u and y to be the radial and tan-gential geostrophic velocities. Overbars denote azi-muthal mean quantities and primes represent departuresfrom the azimuthal mean. Factors to be used in con-verting between nondimensional and dimensional quan-tities are shown in Table 1. The conversion of PV tomks units assumes a near-surface density of 1 kg m23.Note also that T is used to indicate time periods in unitsof the initial eddy turnover time t eddy of the basic statevortex. The eddy turnover time is defined here as t eddy

[ 2pRM/ymax, where RM is the radius of maximum windsand ymax is the tangential wind at that radius. For ourinitial basic-state vortices, t eddy 5 2.91 days.

In the model simulations we use f 5 1024 s21 and H5 10 km. Although the value for f is roughly twicethat found at tropical latitudes, we have chosen the high-er value in order to keep the initial Rossby number Rof the symmetric vortex less than unity. Here R 5 y /fr, where y is the azimuthal-mean tangential wind. Thejustification for using the larger value of f is based onthe idea of incorporating the average rotation rate of anincipient vortex into the definition of f. The physicaljustification behind this idea is provided in Shapiro andMontgomery (1993), where the balance theory derivedwas based on a generalized Rossby number incorpo-rating the local rotation rate of the vortex. An analysisof our results using the primitive equations and asym-metric balance equations (Shapiro and Montgomery1993) is a topic of current work, the results of whichwill be reported in due course. Note that the basic ax-

isymmetrization results described in section 4 have max-imum Rossby numbers of approximately 0.5 with Ross-by numbers of about 0.3 at the RMW. The quasigeos-trophic approximation is used here because of its sim-plicity; we believe that its regime of practical validityextends beyond the range of formal validity R K 1.

In the quasigeostrophic Boussinesq approximation,the PV conservation equation, the invertibility equation,and the thermodynamic equation in nondimensionalform are, respectively,

D Qg 5 0, (4)Dt

2] f2¹ f 1 5 Q 2 1, (5)h 2]z

and

D ]fg 1 w 5 0, (6)Dt ]z

where tildes have been dropped, is the horizontal2¹h

Laplacian operator and Dg/Dt is the material derivativeoperator following the geostrophic wind. Henceforth,only nondimensional variables are used unless specified.The numerical solution of Eqs. (4), (5), and (6) is sum-marized in appendix A.

The fully nonlinear barotropic wavenumber two sim-ulation described in section 4a was performed with asemispectral model based on the two-dimensional non-divergent barotropic vorticity equation on an f plane.Details of this model are given in appendix B.

b. Physical description of convection in thequasigeostrophic model

Although it is obviously not possible to represent con-vective-scale dynamics (e.g., Weisman et al. 1993; Trieret al. 1997) with a quasigeostrophic model, the approachtaken here is a phenomenological one whereby the ver-tical vorticity budget of an ensemble of convective cellsnear or within an incipient vortex is parameterized bypotential vorticity anomalies having a horizontal scaleof approximately 200 km. Neglecting internal friction,Ertel’s PV equation based on dry potential temperatureis

DQ z ·=ua5 ,Dt r

where za is the absolute vorticity, r is the density, andis the heating rate associated with cumulus convection.u

In the quasigeostrophic Boussinesq approximation, onlythe vertical component of is assumed significant,˙z ·=ua

and the density is treated as constant, yielding

DQ ( f 1 z) ]uø . (7)

Dt r ]z

Although the strict quasigeostrophic approximation


TABLE 2. Parameters describing the QG PV anomalies.

Parameter Value

a0

a2

a3

b0

b2

0.9001.001.00

32.0100

b3

xc

yc

xc1

yc1

xc2

yc2

1001.001.001.251.000.751.00

would neglect z compared to f in Eq. (7), we retainboth components of the vorticity for these numericalestimates.

Since realistic convective heating profiles havegreater than zero at low levels and less than zero˙]u/]z

at high levels, convection is seen to create a positivePV anomaly at low levels and a negative PV anomalyaloft. Since is assumed zero at the horizontal bound-uaries (i.e., no explicit enthalpy fluxes at the ocean sur-face), no PV is fluxed into the domain and, consequent-ly, the mass-weighted integral of PV will not change(Hoskins et al. 1985; section 7).

The magnitudes of the convectively induced PVanomalies can be determined if the heating rate isknown. Convective heating rates for midlatitude con-vective systems have been studied both observationallyand theoretically (Gallus and Johnson 1991; Hertenstein1996); an appropriate value of averaged over severaluconvective cells is found to be ø 15 K h21 or 360umax

K d21. From recent radar observations of tropical me-soscale convective systems (Mapes and Houze 1995)one obtains an estimate for of 190–380 K d21. As-usuming further that the heating rate depends on z assin(pz/Hc), where Hc is the vertical scale, and that theincipient vortex has relative vorticity O( f ), we find

6h 2 f u pmaxPV change ø dt9 ø 1-2 PVU,E r Hc0

where one PVU 5 1026 m2 s21 K kg21. Here we haveused f 5 5 3 1025 s21, Hc 5 15 km, and taken r tobe 1.0 kg m23 for a realistic estimate of the magnitudeof PV generation in the lower troposphere. As suggestedby the observations of Zehr (1992) to be discussed insection 5 and displayed in Fig. 16, we integrate for aperiod of 6 h corresponding to the bursts of convectionshown in the figure. Although the peaks in the figurehave a duration of approximately 12 h, we expect thelifetime of the cold cloud tops to be longer than thetime period during which heating is occurring.

The main vortex’s PV can be estimated similarly:2(f 1 z) ]u 2 f u Ntotal 0PV ø ø ø 0.3 PVU.mainvortex r ]z gr

Thus, even being quite conservative, we see that theconvective anomalies for our system can have roughlythe same PV magnitude as the basic-state vortex.

3. Basic states and initialization

a. Introduction

To lay a conceptual foundation for the dynamics ofongoing moist convection in vortex shear flow, we studyfirst the free adjustment or ‘‘relaxation’’ associated withfinite-amplitude asymmetric PV anomalies near the ra-dius of maximum winds of an incipient cyclonic vortex.The combination of a basic-state barotropic circular vor-tex with three different asymmetries is analyzed. The

three representative examples using the initial baro-tropic vortex are a barotropic wavenumber 2 anomalyand three-dimensional two-cluster and single-clusteranomalies that are intended to represent the net effectof tropical convection. The two-cluster and single-clus-ter convective anomalies are defined in section 3c. Forour fourth example we consider the problem of a mid-level vortex in the presence of convection.

b. The basic state vortices

For our first three examples the initial flow consistsof a barotropic circular vortex superposed with eitherof three types of perturbation asymmetries. The initialcircular vortex for the barotropic simulation is the sameas that used in the nondivergent calculations of Mont-gomery and Kallenbach (1997; section 2). The basic-state circular vortex for the three-dimensional model isdefined by the PV field

,2

2b r0 0Q (x, y, z) 5 1 1 a ebasic 0 (8)

where

5 (x 2 xc)2 1 (y 2 yc)2,2r0 (9)

(xc, yc) is the center of the vortex, and parameters suchas a0 and b0 are listed in Table 2. For both cases thebasic-state vortex has an RMW of 0.2 (200 km) and amaximum tangential wind of 0.05 (5.0 m s21). Futurework should investigate the precise thresholds of basic-state vortex strength and areal extent required for thecyclogenesis process described here to occur. The max-imum pressure drop for the basic state in the three-dimensional model is 2.7 mb. Figure 1 shows the cor-responding radial profiles of the azimuthal-mean PV,azimuthal-mean vorticity, azimuthal-mean flow geo-potential, and azimuthal-mean tangential wind for thethree-dimensional model basic state. In the three-di-mensional model, to ensure compatibility with thedoubly periodic boundary conditions, the basic state PVhas been adjusted so that the area integral of the anom-alous basic-state PV is zero. This adjustment is a smalleffect and amounts to the requirement that the net cir-culation is zero on each vertical level.


FIG. 1. Radial profiles of the azimuthal mean potential vorticity,absolute vertical vorticity, flow geopotential, and tangential wind forthe basic-state vortex used in the quasigeostrophic model. To obtainPV in PVU (1026 m2 s21 K kg21), multiply by 0.3. To obtain vorticityin s21, multiply by 1024. To obtain geopotential in J kg21, multiplyby 104. To obtain tangential wind in m s21, multiply by 100. To obtainradial displacement in km, multiply by 1000.

For our fourth example we consider the case of abasic-state vortex that, rather than being barotropic, hasmaximum winds at the middle z level. This configura-tion is of meteorological interest because, as noted insection 1, our initial basic-state vortex could be a me-soscale convectively generated vortex; MCVs typicallyhave maximum tangential winds at midlevels (Johnston1981; Bartels and Maddox 1991). To model such aninitial flow we take

Qmidlevel 5 1 1 sin(pz),2

2b r0 0a e0

where 5 (x 2 xc)2 1 (y 2 yc)2 with the vortex center2r0

(xc, yc), the same as for the barotropic vortex. The pa-rameters a0 and b0 are listed in Table 2 and are thesame as were used for the barotropic basic-state vortex.

c. Initial PV anomalies

For the nondivergent simulation, the basic-state vor-tex is perturbed with a localized wavenumber 2 vorticitydisturbance whose Fourier amplitude is given by

p(r 2 0.070)

20.20z (RMW) sin ,1 20.27 |r 2 RMW| # 0.13;z (r) 52 0, |r 2 RMW| . 0.13.

Here is the azimuthal mean vorticity at thez(RMW)vortex’s RMW and the parameters are chosen to give amaximum asymmetry at the RMW, where the physicalspace asymmetry amplitude is 40% of the basic-stateamplitude. A map plot of the relative vorticity for thisanomaly added to the basic state is shown in the initialcondition of Fig. 2.

The second and third asymmetries considered wereused in the three-dimensional model to simulate the ef-fects of an outbreak of cumulus convection near theinitially circular vortex. A PV anomaly that has ap-proximately the same magnitude as the basic-state vor-tex PV and has the desired property of adding positivePV at low levels and depleting it at upper levels whilekeeping the mass-weighted integral of PV unchanged isgiven by

, (10)2 22b (dr ) 2b (dr )2 1 2 2Q9(x, y, z) 5 a (e 1 e ) cos(pz)2 2

where (dr1)2 5 (x 2 xc1)2 1 (y 2 yc1)2, (dr2)2 5 (x 2xc2)2 1 (y 2 yc2)2, and the parameters a2, b2, xc1, xc2,yc1, and yc2 are listed in Table 2. This configuration,called the two-cluster convective anomaly, has tworegions of convection on opposite sides of the basic-state vortex at radii of 0.25 (250 km) from the centerof the vortex. Contour plots of the PV, vorticity, andflow geopotential for Q2 5 Qbasic 1 are shown inQ92Fig. 3. The PV anomalies are slightly larger at maximumthan the basic-state vortex maximum.1 The maximumpressure drop of each positive anomaly associated with

is 1.0 mb, with a corresponding maximum tangentialQ92wind velocity of 3.0 m s21. The maximum temperaturedeviation from the resting basic state for occurs atQ92z 5 0.5 and is approximately 0.75 K. (The pressure,temperature, and wind fields for the anomalies byQ92themselves are not shown.)

1 Thus, on the 200-km scale the associated heating rate is taken tobe a factor of 3–6 smaller than the observational estimate of section2b.


FIG. 2. Evolution of the nondimensional relative vertical vorticity and the relative vertical vorticityasymmetry for the barotropic wavenumber 2 initial condition. Only the inner 800 km 3 800 kmof the model domain is shown. Time is nondimensional with T 5 1 corresponding to one eddyturnover time (see Table 1 for details).


FIG. 3. Initial nondimensional basic-state vortex 1 two-cluster convective PV, absolute vertical vorticity, and flowgeopotential at top (z 5 1), bottom (z 5 0), and y slice along y 5 1.

As a model for convection that is localized in onearea, we take

22b (dr )3 1Q9 5 a e cos(pz)3 3 (11)

and refer to this configuration as the single-cluster con-vective anomaly. Plots of the PV, vorticity, and flowgeopotential for Q3 5 Qbasic 1 are shown in Fig. 4.Q93The maximum pressure drop of the positive anomalyassociated with is 0.92 mb, with a correspondingQ93maximum tangential wind of 2.8 m s21. The maximumtemperature deviation from a resting basic state for the

anomaly occurs at z 5 0.5 and is approximately 0.75Q93K. (The pressure, temperature, and wind fields for the

anomaly are not shown.)Q93When the negative anomalies associated with andQ92are added to the basic-state PV the total PV remainsQ93

positive, thereby ensuring symmetric stability ( fQ . 0)of the model atmosphere.

For the midlevel vortex example we used the samesingle-cluster convective anomaly as that describedabove. Thus the PV distribution for this example is giv-en by Q4 5 Qmidlevel 1 .Q93

4. Fundamentals of three-dimensional vortexaxisymmetrization

a. The barotropic wavenumber two asymmetry

To validate the wave–mean flow predictions of Mont-gomery and Kallenbach (1997), as well as to provide aconceptual foundation for the upcoming baroclinic ex-periments, we consider first the relaxation of the initialbarotropic wavenumber 2 anomaly. Figure 2 shows theevolution of the total and asymmetric relative vorticityfor this case. Waves propagating both azimuthally andradially are clearly evident in the asymmetric vorticity.


FIG. 4. Initial nondimensional basic-state vortex 1 single-cluster convective PV, absolute vertical vorticity, andflow geopotential at top (z 5 1), bottom (z 5 0), and y slice along y 5 1.

In particular, as the positive and negative vorticity per-turbations are sheared into trailing spirals by the meanvortex the perturbations propagate outward. Excitationof secondary wave features in the interior of the vortexfollowing the shearing of the initial asymmetries is alsoevident.

The axisymmetrizing wave disturbances evident inFig. 2 cannot be gravity waves since gravity waves arecompletely excised in the nondivergent model. Indeed,the waves are vortex Rossby waves whose restoringmechanism is associated with the radial gradient of ba-sic-state vortex vorticity. The basic theory for thesewaves was developed by Montgomery and Kallenbach(1997). The radial group velocity for vortex Rossbywave packets in the WKB approximation is given by

22knz 90C 5 ,gr 22n2R k 10 21 2R0

where n is the azimuthal wavenumber and is thez90basic-state radial vorticity gradient at reference radiusr 5 R0. The radial wavenumber k is given by k(t) 5 k0

2 , where k0 is its initial value, t is time, andntV90is the radial gradient of the basic-state angular ve-V90

locity. Outside the RMW, our initially circularly sym-metric anomaly will be deformed to a downshear-tiltedpatch. For downshear tilt, k . 0. Since for our vortex

, 0, k will grow more positive with time. If theV90relative vorticity gradient is negative, as is initially thecase, Cgr will be positive and wave packets will prop-agate outward. In the upcoming examples using larger-amplitude initial asymmetries, the outermost wave cre-ates a region of positive mean vorticity gradient; in thatregion, Cgr , 0 and wave packets propagate inward.

The solid curves in Fig. 5 show the wave-inducedchanges in azimuthal mean relative vorticity and tan-gential velocity predicted by the nonlinear semispectralmodel. The dashed lines show rhe corresponding quasi-


FIG. 5. Change in the relative vertical vorticity and mean tangential wind at T 5 4 for thebarotropic wavenumber two system. Here, dy in m s21 is obtained by multiplying by 100. Radialdisplacement in km is obtained by multiplying by 1000. Solid line shows the result of the fullynonlinear simulation and the dashed line shows the quasi-linear prediction.

linear prediction obtained by using just the linear so-lution to evaluate the eddy vorticity flux. Overall, theagreement is quite good. A notable feature of Fig. 5 isthe acceleration or spinup of the mean tangential windsand mean vorticity near the radius of the initial asym-metry. This spinup was predicted by Montgomery andKallenbach (1997); as we will see, it persists in thethree-dimensional setting.

Although significant excitation of higher wavenum-ber components does occur, analysis of the wavenumber2 vorticity shows that the instantaneous wavenumber 2packet propagates radially outward to a stagnation ra-dius (r 5 0.27) close to the zero in the d y plot. Thisqualitative behavior was predicted in the WKB frame-work of Montgomery and Kallenbach (1997) and is con-sistent with the wave activity interpretation of Held andPhillips (1987) generalized to vortex flow. Quantita-tively, the quasilinear prediction for d y max is 0.10 ms21; the value observed in the nonlinear model is 0.09m s21. The Montgomery and Kallenbach (1997) results,valid at second order in the wave amplitude, are stillapproximately valid. Thus the interaction of vortexRossby waves with the mean flow captures the essenceof the physics at these small but finite amplitudes. Athigher amplitudes we find that wave–wave interactionsplay a more significant role, but the structure of dy anddz are nevertheless observed to be qualitatively similarto their small amplitude counterparts (e.g., see Fig. 9).

As the initial asymmetry is sheared by the mean vor-tex, some of the positive vorticity anomaly is transportedtoward the interior of the vortex. The remaining portionof the positive vorticity anomaly is transported outwardto form the positive vorticity filaments that orbit thevortex core. The negative vorticity anomaly is alsotransported outward and becomes more nearly axisym-metric. Lagrangian trajectory analyses (see appendix C)lend further support to these ideas about vorticity trans-

port in the symmetrization process. A contour plot (notshown) of the initial azimuthal-mean vorticity subtract-ed from the total vorticity at T 5 2.5 exhibits centralregions of vorticity augmentation surrounded by spiralarms of vorticity depletion, with an outer region of vor-ticity augmentation as well. The authors studied La-grangian back-trajectories for particles destined to befound in each of these regions. Particles in the innervorticity augmentation region were found to originatein the initial positive vorticity anomalies, in agreementwith our understanding that positive vorticity from theanomalies moves inward and pools near the center ofthe vortex. Particles in the vorticity depletion regionbacktracked to the initial vorticity depletion regions ofthe wavenumber 2 anomaly.

A nearly neutral discrete vortex Rossby mode asso-ciated with a sign change of the radial vorticity gradientis visible in the barotropic experiment for T $ 4. Suchmodes are discussed in section 4g.

b. The two-cluster convective anomaly

We consider next the two-cluster convective anomaly.Figure 6 shows the potential vorticity field for top, mid-dle, and bottom z surfaces as a function of time. At thetop level, the negative anomalies are expelled from themain vortex and are advected around the vortex. Thenegative anomalies are not assimilated by the vortex butremain intact. At low z levels the behavior is similar tothe barotropic case discussed above; the positive anom-alies merge into the main vortex, with accompanyingproduction of high-PV filaments, and again the vortexsymmetrizes almost completely. However, due to thewave-induced change in the mean vorticity gradientshown below, neutral or weakly unstable modes are ev-ident in the inner core region (see section 4g).

Figure 7 shows the azimuthal mean and asymmetric


FIG. 6. PV evolution for the two-cluster convective case at top, middle, and bottom model levels.

PV as a function of time at the highest model level (z5 1). Disturbances to the mean flow are evident. Asnoted earlier, the PV deficit regions move outward. Ax-isymmetrization does not occur at upper levels. The flowthere resembles a vortex tripole (Polvani and Carton1990; Orlandi and van Heijst 1992). From a linear waveviewpoint the tripole can be thought of as a finite am-plitude generalization of a wavenumber 2 discrete orweakly unstable vortex Rossby mode that is supportedby a sign change in the mean radial PV gradient (seesection 4g).

The bottom level (Fig. 8) shows an increase in meanPV at a radius of approximately 0.1 (100 km), especiallyat earlier times. As in the barotropic case, there is anoverall steepening of the PV gradient. Figures 9a,b showthe changes in the azimuthally averaged PV and tan-gential velocity at the bottom z surface. The similarityto the barotropic example (Fig. 5) is striking. The pool-ing of high PV at radius 0.125 (125 km) is evident. Aspinup of 1.0 m s21 occurs over the period of 1t eddy

(approximately 3 days). No further acceleration is ob-served after 1t eddy.

In the double-cluster simulation, the stagnation radiusof the initial wavenumber 2 wave packet is again ap-proximately equal to the radius of the zero in the d yplot (r 5 0.3). Taking the primary azimuthal mode andits next two harmonics into account, the value of d y max

estimated by the quasi-linear nondivergent model is 0.94m s21.

Forward Lagrangian trajectories for the two-clusterconvective case at the lowest model level have also beenanalyzed. Particles were placed randomly on a circle ofradius 0.1 (approximately the size at half-maximum ofthe positive anomaly) centered at the anomaly center.The results for a few selected particles are shown inFig. 9(c). Many of the particles move inward toward r5 0.125 (125 km). The others move outward to formthe outer high-PV filaments. Trajectories at the highestlevel (not shown) were also analyzed. At this level,particles from the negative anomalies move generally


FIG. 7. Evolution of the azimuthal mean PV and asymmetric PV for the two-cluster convective anomaly at thehighest model level (z 5 1). Contour interval is fixed at 0.1 for all plots. One nondimensional unit of radial displacementequals 1000 km.


FIG. 8. Evolution of the azimuthal mean PV and asymmetric PV for the two-cluster convective anomaly at thelowest model level (z 5 0). One nondimensional unit of radial displacement equals 1000 km.


FIG. 9. (a) Change in azimuthal mean PV (dQ) and (b) change in mean tangential wind (dy )at T 5 2.5 for the two-cluster convective case at the lowest model level (z 5 0). Here, dy inm s21 is obtained by multiplying by 100. Radial displacement in km is obtained by multiplyingby 1000. (c) Forward trajectories for the two-cluster convective anomaly at the lowest modellevel (z 5 0). The dotted line shows the location of the anomaly from which the particlesoriginate. Only the inner 1200 km 3 1200 km of the model domain is shown.

outward, in agreement with Fig. 7. Particles from thepositive anomalies either move inward to form the slightbump of positive mean PV shown at r ø 0.225 in theradial profiles of Fig. 7, or move outward to form thesecondary maximum at r ø 0.45.

The Lorenz energy cycle (Holton 1992) for the two-cluster convective experiment has also been examined(Enagonio and Montgomery 1998). As expected, thelargest energy storage is found to be in the mean-flowkinetic energy; the largest energy conversion is thatfrom eddy kinetic energy to mean kinetic energy, cor-responding to the spinup of the basic-state vortex by theeddies observed with the wave–mean flow diagnostics.

c. The single-cluster convective anomaly

For our third example we consider the single-clusterconvective anomaly. The single-cluster configuration isintended to simulate a localized outbreak of convectionnear the preexisting vortex. For this case the center ofthe (r, l) coordinate system is taken to be the instan-taneous geopotential minimum on the lowest level. Thebehavior is in many ways similar to the two-clusterconvective anomaly. At the top level, the positive andnegative anomalies move apart; at the lowest level thetwo positive anomalies move together and symmetrize.The PV evolution for the single-cluster case at the bot-tom level is shown in Fig. 10.


FIG. 10. Evolution of the azimuthal mean PV and asymmetric PV for the single-cluster convective anomaly at thelowest model level (z 5 0). Contour interval is fixed at 0.1 for all plots. One nondimensional unit of radial displacementequals 1000 km. Radial profiles extend only to approximately 700 km because of the motion of the lowest-levelcenter of the system.


FIG. 11. (a) Change in azimuthal mean PV (dQ) and (b) change in azimuthal mean-tangentialwind (dy ) at T 5 2.5 for the single-cluster convective case at the lowest model level (z 5 0).Here, dy in m s21 is obtained by multiplying by 100. Radial displacement in km is obtained bymultiplying by 1000. (c) Forward trajectories for the single-cluster convective case at the lowest-model level (z 5 0). The dotted line shows the location of the anomaly from which the particlesoriginate. Only the inner 800 km 3 800 km of the model domain is shown.

The lowest-level dQ and d y distributions, plotted inFigs. 11a,b, show a new phenomenon: transport of PVto the center of the vortex. This occurs because thesingle-cluster convective anomaly has a wavenumber 1Fourier component; only wavenumber 1 can transportparticles to the center of the vortex. The Lagrangiantrajectories show the same effect. Figure 11c shows theforward trajectory paths for a few selected particlesoriginating on the 1PV anomaly for the single-clustercase. Unlike Fig. 9c, in which particles are excludedfrom the center of the vortex, in Fig. 11c particles passarbitrarily close to the center.

Figure 11b shows a low-level spinup of about 0.4 ms21 for single-cluster convection. This is about a factor

of 2.5 smaller than for the two-cluster convective anom-aly. This reflects the fact that the total initial forcing inthe single-cluster case is approximately a factor of 2smaller: we use the same magnitude anomaly in bothcases, but there are two positive anomalies for the two-cluster case and only one for the one-cluster example.For a comparison of the spinup due to single-clusterversus double-cluster anomalies as a function of theanomaly amplitude, see section 4f.

Table 3 shows the dependence of the final surfacemaximum azimuthal mean-tangential wind speed aboutthe geopotential minimum on the radial location of theconvective patch for the single-cluster convective con-figuration. Convection at or near the center of the basic-


TABLE 3. Final surface maximum azimuthal mean-tangential windas a function of the radial location of the convective patch from theinitial vortex center.

Radius ofconvective anomaly

(km)maxy

(m s21)

0100250400500

7.06.45.64.94.7

state vortex produces significantly greater values ofy max. For radii much greater than the initial radius ofmaximum winds of the basic-state vortex, a slight spin-down (d y max , 0) is observed. The final radius of max-imum winds also depends on the initial radial locationof the convection, ranging from approximately 150 kmfor convection at the center to 235 km for convectionat 500-km radius.

The largest increase in tangential velocity occurswhen the convective patch is placed at the center of theinitial basic-state vortex. In view of this fact, the readermay wonder why our emphasis is on the role of asym-metries in the spinup process. However, it should benoted that, assuming that at the genesis stage the pre-existing vortex does not play a strong role in organizingthe convection, it is more probable that convective out-breaks will occur somewhere near the periphery of thevortex rather than very close to the center; thus webelieve that asymmetric processes are likely to be animportant contribution to mechanisms for tropical cy-clogenesis. As discussed in section 6, observations ap-pear to indicate that asymmetries play an important rolein the genesis process. When symmetric convection doesoccur, it can be expected to yield stronger and morerapid spinup.

d. A midlevel vortex with convection

The time evolution of the midlevel vortex configu-ration is summarized in Figs. 12 and 13. The figuresshow contours of PV on z 5 0, z 5 0.25, z 5 0.5, z 50.75, and z 5 1, as well as contours of PV versus x andy on z 5 0. The low-level PV anomaly, initially at 250km from the center of the basic state vortex, is drawnapproximately 200 km into the center to a position un-derneath the main vortex and subsequently remainsaligned with it, producing a vertically stacked vortex inthe mid- to low-level troposphere. In contrast, the neg-ative PV anomaly at upper levels is expelled laterallyfrom the axis of the now-aligned vortex system. Thisexample demonstrates that our axisymmetrizationmechanism could cause an MCV to appear to ‘‘builddownward’’ in the presence of peripheral convection,resulting in spinup of strong winds at the surface.

The midlevel vortex experiment shows some spinupof the azimuthal-mean tangential wind about the center

of the midlevel vortex. At T 5 0, the mean-tangentialwind maximum occurs at about 300 km from the axisof the mean vortex, approximately corresponding to theposition of the convective anomaly. The surface meantangential wind maximum measured about the axis ofthe midlevel vortex at the surface is 2.3 m s21. At T 52.5, the RMW at the surface has contracted to 150 km,and the mean-tangential wind maximum has increasedto 3.0 m s21, with a maximum spinup of 1.7 m s21. Thesurface vortex is now capable of further growth if con-vection continues to occur around it.

e. Warm core formation through axisymmetrization

In the course of the two-cluster relaxation experiment,a slight warming of 0.08 K in the azimuthal mean-tem-perature field is observed at mid- to low levels. As wewill see in section 5, in the presence of ongoing con-vection a strong warm core of magnitude 3–5 K formsduring the axisymmetrization process. We now discussthe physical basis for the warming observed in the re-laxation experiments.

On azimuthally averaging the thermodynamic equa-tion, we obtain

] ]f ] ]f9 g25 2 ru9 2 N w 1 u . (12)[ ]]t ]z r]r ]z u0

Contributions to ]f z/]t come from radial eddy-heat flux,mean vertical motion, and heating due to convection. Inthe relaxation experiments after the initial pulse 5 0.¯u

The mean vertical motion is given by the radial de-rivative of the transverse streamfunction:

]w 5 c ,

r]r

where c is deduced upon solving the Sawyer–Eliassenequation

2 2] ] f ] c ]E ]F2N c 1 5 2 , (13)

21 2]r r]r r ]z ]r ]z

with

] ]f9E 5 2 ru9 ,1 2r]r ]z

and

f ]F 5 2 [r(ry9u9)].

2r ]r

Note that Eq. (13) is dimensional. Here, E representsthe contribution from eddy-heat flux forcing and F rep-resents the contribution from eddy-momentum flux forc-ing.

Neglecting for the moment the contribution from theeddy-heat fluxes, the physics responsible for the meanvertical motion can be illustrated with a simple example.We consider the change in the mean tangential wind d y


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induced by eddy-momentum flux forcing following theimposition of a convective PV anomaly at low levelsnear the undisturbed circular vortex. As we have alreadyseen in section 4a, a local torque is exerted on the meancircular vortex during the axisymmetrization processthat causes a net spinup of the tangential winds. Becausethe secondary circulation always acts to oppose changesinduced by the geostrophic flow, the mean-transversecirculation consists of a radially outward flow near thed y maximum that tends to resist the spinup. Similarly,near the d y minimum the mean-transverse circulationis radially inward. The resultant mean convergence leadsto mean ascent in between the two regions; by massconservation, mean subsidence develops near the centerof the vortex. When the contribution from eddy-heatfluxes is included the dynamics is no longer so simple.

As a first look at the relative importance of heat andmomentum flux contributions to the development of thewarm core, azimuthal mean Eliassen–Palm (EP) fluxvectors (Edmon et al. 1980) for the two-cluster relax-ation experiment have been analyzed (Enagonio andMontgomery 1998). In cylindrical coordinates, the qua-sigeostrophic EP flux vector is defined by

f ]f9F 5 2ry9u9 , 2 ru9 .

21 2N ]z

A plot of the EP flux vectors along with contours ofthe radial PV flux (u9Q9 5 2= ·F/r) for the two-clusterrelaxation experiment at time T 5 0.5t eddy shows thatat low levels for r , 0.3, u9Q9 is negative, indicatingthat the eddy PV flux is upgradient (into the vortex);for r . 0.3, u9Q9 is positive, showing the downgradientflux of eddy PV into the high-PV filaments at largerradii. The plot also indicates that at this time the heatflux is into the vortex at upper levels, out of the vortexat midlevels, and negligible at low levels. At low levels(z , 0.4) the cyclonic momentum flux is predominantlyinward, and greater values of = ·F here agree with ourother findings that the energy transferred to the meanflow is greater at lower levels.

As indicated in Eq. (13), the mean-transverse stream-function is derived from derivatives of the fluxes ratherthan from the fluxes themselves. For a direct comparisonof the effects of momentum flux to heat flux we turnto explicit computation of the forcing terms in the Saw-yer–Eliassen equation. A simulation initialized with thetwo-cluster convective anomaly was studied with re-sultant fields output every 0.125t eddy in order to studythe rapid azimuthal shearing of the asymmetry. At eachoutput time the two forcing terms ]E/]r and 2]F/]zwere calculated and plotted. The results for some se-lected times are shown in Fig. 14. We see from Fig. 14that at early times the momentum flux term dominatesthe heat flux term everywhere. The effect of the mo-mentum flux is best seen by looking at 2]F/]z at T 50.375 in Fig. 14. The momentum flux forcing at lowlevels inside the radius of maximum winds is strongly

positive. The elliptic operator in Eq. (13) tends to re-verse the sign of the operand, so that in the region where2]F/]z is strongly positive, c is strongly negative. Itfollows that w is negative near the center of the vortexat low levels; thus there is subsidence in that region asexpected.

All throughout this process the heat flux term is non-zero; by T 5 0.375 the heat flux becomes a significantcontribution to the forcing at upper levels. Nevertheless,the maximum warming occurs in the mid- to lower lev-els where the momentum flux is dominant.

f. Nonlinear feedback in the axisymmetrizationprocess

As will be evident in the section on applications totropical cyclogenesis, it is of interest to study the de-pendence of the maximum spinup d y max on the asym-metry amplitude. In the quasi-linear nondivergent re-gime d y scales as h2ymax, where ymax is the maximumtangential wind of the basic-state vortex and h is thestrength of the asymmetry relative to the basic-statevortex (Montgomery and Kallenbach 1997). Using h ;Q9/Qmax, where Q9 is the maximum of the PV anomalyand Qmax is the maximum PV of the basic state, andrecalling that the invertibility relation is linear, we findthat d y ; . In our experiments Q9/Qmax is not2Q9 /ymax

small, but this scaling may nevertheless be approxi-mately valid. Experiments were performed to test thedependence of d y max and the maximum temperaturechange d Tmax on Q9 and ymax.

Figure 15 shows the results of these tests, along withthe predicted maximum d y and maximum dT assuminglinear and quadratic scaling in Q9. Note that, since theupper levels do not become axisymmetric, one obtainssmall fluctuations in the azimuthal mean quantities forthe single-cluster experiments as the center of the systemmoves. For the two-cluster case at the tested amplitudes,the scaling of d y max is intermediate between linear andquadratic. However, the temperature change dTmax hasa greater than quadratic dependence on Q9. For the sin-gle-cluster case, the dependence of d y max is greater thanquadratic, but the variation of the temperature changewith amplitude is quite small.

The dependence of d y max and dTmax on ymax was alsotested. For the double-cluster experiment, d y max is foundto be approximately inversely proportional to ymax foran increase of 20% in ymax. For the single-cluster case,an increase of 20% in ymax results in a 24% decrease ind y max. It is interesting to note that for both cases, dTmax

is essentially unaffected by a change in vortex strength.We have not yet developed a scaling argument that pre-dicts the dependence of dTmax on anomaly amplitudeand vortex strength.

These results are indicative of nonlinear feedback inthe vortex dynamics. From the dimensional vertical vor-ticity equation for quasigeostrophic dynamics,


FIG. 14. Momentum flux and heat flux forcing terms for the Sawyer–Eliassen equation as a function of time for the two-clusterconvective relaxation experiment. To obtain the forcing terms in s23, multiply the nondimensional quantity by 10213.


FIG. 15. Maximum dy and maximum change in temperature as a function of anomaly amplitude for the double-cluster and single-clusterrelaxation experiments. Model results are compared with predictions assuming linear and quadratic scaling. The amplitude is displayed inunits of the nominal anomaly pulse amplitude. Here, dy is in m s21 anddT in K.

]z 1 ] f ]g 5 2 (ru9z9) 2 (ru ), (14)g a]t r ]r r ]r

some type of nonlinear spinup effect is expected.On the face of it one might claim that the spinup in

these experiments occurs simply because the azimuthal

mean-radial secondary circulation is converging theconvectively generated relative vorticity into the storm’sinner core. This idea is incorrect for two reasons. First,as seen from Eq. (14), in quasigeostrophic theory theazimuthal mean circulation (which is solely ageostroph-ic) converges only planetary, not relative vorticity. Sec-


FIG. 16. Example of a profile of convective activity for a tropicalcyclone. The plot shows a time series of the percentage of the areaof a tropical cloud cluster with brightness temperature TB less than2658C, indicative of deep cumulus convection (see section 5a). Ty-phoon Abby, 1983 from Zehr (1992).

ond, most of the spinup occurs in the inner core of thestorm where the azimuthal mean-radial circulation atlow levels is outward (hence divergent). From the PVviewpoint, since potential vorticity is advected solelyby the geostrophic wind, its advection into the coreoccurs purely through eddy transports.

g. Amplitude sensitivity and wave-induced discretevortex modes

In all the experiments described thus far, the upper-level PV anomalies do not become axisymmetric. Inaddition to the fact that the upper-level anomalies existin favorable shear, we believe that the disruption ofaxisymmetrization at upper levels can be traced to theexistence of discrete neutral or unstable vortex Rossbymodes propagating azimuthally around the vortex cen-ter. For linear waves neutral nonsingular modes requirea vanishing radial PV gradient somewhere in the flow(Pedlosky 1987, section 7.8), while unstable modes re-quire ]q /]r to change sign at least once (Gent andMcWilliams 1986). To investigate these ideas, we stud-ied smaller-amplitude disturbances that permitted us toenter a quasi-linear regime.

When the amplitude of the anomalies was decreasedby a factor of 5 from the nominal case, we still obtaineda weak wave-induced sign change of the azimuthal ra-dial PV gradient at upper levels near r 5 350 km. Underthese conditions one can discern a persistent upper-levelwavenumber 2 disturbance after the axisymmetrizationat low levels is complete. Despite the presence of dif-ferential rotation at all levels, the disturbance retainedits shape as it propagated cyclonically around the vortex.

In order to verify the fact that this mode’s existencedepended on the sign change of the azimuthal meanradial PV gradient, we moved the anomalies inward toa position 150 km from the center of the vortex. Thischange had the effect of superposing the anomalies ona larger value of the basic-state PV so that the basic-state ]q /]r dominated the wave-induced radial PV gra-dient. In this configuration we observed axisymmetri-zation at all levels, verifying the disappearance of thediscrete or weakly unstable baroclinic modes.

The existence of wave-induced discrete neutral orweakly unstable baroclinic vortex modes raises intrigu-ing possibilities about their ability to orchestrate furtherconvective bursts near the RMW and sustain intensifi-cation. A thorough investigation of these ideas requiresa self-consistent cumulus convection model, which isbeyond the scope of this paper. This topic, as well asfurther investigation of the underlying dynamics of thediscrete vortex Rossby modes, remains for future work.

5. Three-dimensional vortex dynamics underconvective forcing: Application to tropicalcyclogenesis

a. Pulsed convective studiesIn the previous section, we described studies of the

axisymmetrization of a vortex forced by initial asym-

metric PV anomalies. We examined various aspects ofthis problem and demonstrated that the PV anomaliescause vortex spinup.

For a more realistic model of cyclogenesis, we shouldincorporate the fact that convection is not simply aninitial forcing condition on the vortex; convective ac-tivity is often ongoing. Figure 16 (from Zehr 1992)shows a time series of convective activity in a tropicalcloud cluster that eventually became Typhoon Abby in1983. The level of convection is indicated by the frac-tion of the cloud cluster area with IR brightness tem-perature TB less than 2658C, indicative of deep cumulusconvection. Pertinent to our work is the occurrence ofmultiple bursts of convection, at intervals of approxi-mately 24 h, prior to the tropical depression designation.

In order to simulate this multiple-burst effect, ‘‘puls-es’’ of convective activity in the form of small-scale PVanomalies were added to the PV field at intervals (typ-ically 0.5t eddy, half the initial eddy turnover time of thevortex, or about 1.5 days) during the time stepping pro-cess. A pulse consisted of a PV anomaly processing thesame shape as the initial convective anomaly. Althoughthe convective bursts shown in Fig. 16 are of extendedduration, this feature is impractical to simulate in ourmodel. Instead, our PV pulses occur all at once at thechosen time step. In the quasigeostrophic model, thewind and height fields adjust instantaneously to the PVfield in accord with the invertibility relation [Eq. (5)].Recall that gravity–inertia waves are excised in the bal-anced model. Model runs thus consisted of an initialvortex and asymmetric PV anomaly, which were pulsedwith additional asymmetric PV anomalies each 0.5t eddy.The pulse was applied just after the model fields were


written out. Since the runs were 2.5t eddy in duration,there were typically four pulses in addition to the initialanomaly. Because the shear time (see appendix A) t shear

5 0.2t eddy, applying a pulse each 0.5t eddy just after thedata was written out assured that the mean fields hadstabilized subsequent to the pulse by the next outputtime.

Since the PV pulses had a z dependence of cos pz,as much negative as positive PV was added within thedomain by each pulse, ensuring that the total PV of thefluid did not change. This is consistent with the generalrequirement noted in section 2b that, in the absence offriction and heat fluxes on boundaries, convection mere-ly redistributes PV such that its mass-weighted integralis invariant.

In most of the simulations described here, the pulsesgrew in amplitude, with the amplitude Ak of the kth pulsegiven by

Ak 5 (1 1 e)kA0, (15)

where A0 is the initial pulse amplitude and e 5 0.2. Theamplitude of the convective maximum was increased tosimulate the increasing relative vorticity being con-verged; a primitive equation effect not captured in thestrict implementation of our quasigeostrophic model,which only converges planetary vorticity explicitly. Thetime rate of change of the potential vorticity is givenby Eq. (7); thus the amplitude of the first pulse is givenby A1 5 (1 1 e)A0, where e represents the contributionof the relative vorticity; similarly A2 5 (1 1 e)A1 5 (11 e)2A0, and so forth. As we will see below, if the pulseamplitude does not increase, considerably less spinupis achieved.

Figure 17 shows the azimuthal mean PV fields andazimuthal mean tangential winds at the lowest level asa function of time for a pulsed experiment with a two-cluster convective anomaly, with pulse amplitude in-creasing according to Eq. (15). The vortex tangentialwinds increase by 10 m s21, yielding a final maximumtangential wind of 16 m s21.

Figure 18 shows the azimuthal mean potential vor-ticity, flow potential temperature, flow geopotential, andtangential wind plotted as a function of r and z at T 52.5 for the two-cluster pulsed asymmetry. Anticyclonicflow at upper levels is evident. Another feature of in-terest is the strong warm core (ø 5 K) that forms at thecenter of the vortex (one nondimensional temperatureunit corresponds to 30 K). In section 5c the formationof the warm core is discussed in greater detail.

For comparison, an experiment in which the pulseamplitude remained fixed with time [e 5 0 in Eq. (15)]was also performed. Because the successive pulses offixed PV tend to contribute a fixed value of PV to thevortex, in this case less spinup was obtained. Since thefluid is incompressible, particles can replace but notoverlap each other. Thus as the high PV particles areattracted toward the center, they tend to pile up at the

outside radius of the inner PV maximum, leading to abroadening of the inner PV maximum with time.

An experiment with single-cluster pulsed convectiveforcing was also performed. In this experiment a spinupof approximately 7.0 m s21 was obtained. As in theunpulsed axisymmetrization experiments, the single-cluster anomaly gives less spinup than the two-clusteranomaly. If the low-level spinup scaled linearly withthe amount of positive PV injected, the single-clusteranomaly would be expected to contribute only 50% asmuch PV to the vortex as the double cluster does. Ac-tually, it contributes considerably more than 50%; thesingle-cluster anomaly appears to be more efficient atthe spinup process. This greater efficiency is likely amanifestation of the nonlinear feedback effect discussedin section 4f.

Plots of the time evolution of the mean and asym-metric PV fields on the highest and lowest levels forthe pulsed single-cluster asymmetry show that at lowto midlevels the positive anomaly is pulled into the mainvortex during the symmetrization process. At high levelsthe negative anomaly is repelled from the main vortex;the anomaly rotates around the parent vortex and littleor no axisymmetrization occurs.

Figure 19 shows r–z mean fields for the pulsed ex-periment with the single-cluster convective anomaly. Anupper-level anticyclone and warm core are evident inthis figure.

b. Sensitivity tests

Recognizing the chaotic nature of cumulus convec-tion, it is important to demonstrate that the hypothesisproposed here is not sensitive to details of how con-vection is represented in our model. Strictly speaking,this problem should be studied with a full physics modelcapable of representing cumulus convection and me-soscale dymanics. Such an analysis is beyond the scopeof the present study. As a substitute, we have performedsome simple sensitivity tests with the quasigeostrophicmodel, which assure that our results are essentially in-dependent of details of the pulsing scheme such as thepulse frequency and location. A summary of the sen-sitivity tests is given in Table 4.

The first of these tests showed that our results areactually quite sensitive to the radial location of the con-vection. We performed a simulation that was the sameas the nominal two-cluster experiment except that thetwo convective anomalies were placed at 400 km, ratherthan 250 km, from the center of the vortex. The pulseamplitude in this run was increased according to Eq.(15). For the 400-km case, the tangential wind maximumbroadened relative to the 250-km case. The maximumspinup, 4.9 m s21, occurred at a radius of approximately350 km. These results are qualitatively consistent withthe results for the relaxation experiments described insection 4c.

To test for sensitivity to the pulse location when the


FIG. 17. Time evolution of the azimuthal-mean PV field (Q) and azimuthal-mean tangential wind(y ) at z 5 0 for a pulsed two-cluster PV asymmetry. To obtain the tangential wind in m s21, multiplythe nondimensional velocity by 100. To obtain the radial displacement in km, multiply the non-dimensional quantity by 1000.


FIG. 18. Azimuthal-mean total potential vorticity Q(r, z), flow potential temperature u(r, z), flowgeopotential f (r, z) and tangential wind y (r, z) for the pulsed two-cluster convective anomaly,contoured as a function of r and z, at T 5 2.5. To obtain potential vorticity in PVU (1026 m2 s21 Kkg21), multiply the nondimensional quantity by 0.3, where a mean-tropospheric density of 1.0 kgm23 has been assumed. To obtain temperatures in K, multiply the nondimensional quantity by 30.To obtain geopotential in J kg21, multiply the nondimensional quantity by 104. To obtain velocitiesin m s21, multiply the nondimensional quantity by 100. To obtain radial displacement in km, multiplythe nondimensional quantity by 1000.

pulse anomalies were at the same radius from the vortexbut had different orientations, we created what we referto as ‘‘scrambled’’ pulses. The initial two-cluster con-vective anomalies were located at 250 km along the xaxis, as usual. However, the first secondary pulse anom-alies, though also in the two-cluster configuration, wereplaced at 250 km along the y axis, rotated 908 from theinitial anomaly. The second pulse was rotated back 908to the position of the initial anomaly, and thereafter thepulses alternated between these two orientations alongthe x and y axes. The scrambled pulse runs gave quitesimilar results to the normal runs, with a maximum spi-nup of 10.3 m s21.

The dependence of the spinup on the pulse frequencywas also tested. The vortex’s shear time is 5.6 3 104 s,while the eddy turnover time t eddy is 2.5 3 105 s. Sincet shear is considerably less than t eddy and we normallypulse the system every 0.5t eddy, the pulsed asymmetriestend to be sheared well before the onset of the next

pulse. Thus, we expect that for a fixed total number ofconvective pulses, the mean tangential winds and thusthe spinup should be approximately independent of thepulse frequency. To check this, we performed experi-ments for both the single- and the double-cluster caseswith the pulse frequency doubled, but with the sametotal number of pulses as in the nominal runs. Theseexperiments correspond to all of the spinup occurringin approximately 3.6 days rather than approximately 7.3days. The values of maximum d y differ by about 1–2m s21 from the lower-frequency values. Thus, as ex-pected, the frequency of the convective pulsing in thisfrequency range does not significantly affect our results.In fact, if the convective anomalies were stronger, thespinup time could be considerably shorter. It should berecalled that the amplitudes used for the PV anomaliesare quite conservative, given our estimate of tropicalconvective heating rates in section 2b. In the relaxationexperiments, we find that axisymmetrization at middle


FIG. 19. Azimuthal mean-total potential vorticity Q(r, z), flow potential temperature u(r, z), flowgeopotential f (r, z), and tangential wind y (r, z) for the pulsed single-cluster convective anomaly,contoured as a function of r and z, at T 5 2.5. To obtain potential vorticity in PVU (1026 m2 s21

K kg21), multiply the nondimensional quantity by 0.3, where a mean-tropospheric density of 1.0 kgm23 has been assumed. To obtain temperatures in K, multiply the nondimensional quantity by 30.To obtain geopotential in J kg21, multiply the nondimensional quantity by 104. To obtain velocitiesin m s21, multiply the nondimensional quantity by 100. To obtain radial displacement in km, multiplythe nondimensional quantity by 1000.

TABLE 4. Sensitivity tests of the quasigeostrophic model underconvective forcing.

Run type Max. d (m s21)y

Nominal 2-cluster convective pulsed2-cluster, constant pulse amplitude2-cluster, anomalies at 400 km2-cluster, scrambled2-cluster, high frequencyNominal 1-cluster convective pulsed1-cluster high frequency

10.26.44.9

10.39.47.08.9

to low levels occurs even when the PV anomaly am-plitude is increased by a factor of 3 over its nominalvalue. The maximum spin up in this experiment is 7 ms21, yielding a final y max of 11.5 m s21.

c. Warm core formation

Figure 20 shows the time evolution of the azimuthalmean-potential temperature deviation from the restingatmosphere for the nominal two-cluster pulsed convec-tive anomaly. As before, one nondimensional temper-ature unit corresponds to 30 K. The azimuthal meanvortex has an initial warm core of approximately 0.5 Kat z 5 0.5. The warm anomaly arises from the wave-number 0 component of the imposed asymmetry.

In the course of the experiment, which correspondsto 2.5t eddy or about 7.3 days, the warm core intensifiesto a 5 K level. As demonstrated above, our model can

actually achieve full spinup in 1.25t eddy or about 3.6days. Thus we can demonstrate the buildup of a 5 Kwarm core on a reasonable timescale for tropical cy-clogenesis. This is a significant result, as it links theformation of a warm core to the axisymmetrization pro-cess.


FIG. 20. Time evolution of the azimuthal mean-potential temperature deviation from the resting atmosphere for the nominalpulsed two-cluster convective experiment. To obtain temperature in K, multiply the nondimensional quantity by 30. To obtainradial displacement in km, multiply the nondimensional quantity by 1000.

To investigate the mechanisms responsible for thewarm core structure under multiple pulsing, we considerfirst the azimuthal mean-vertical velocity deduced fromthe thermodynamic Eq. (6). Normally, in multiple-pulsed runs the pulsing occurs each 0.5t eddy, just afterthe model fields and azimuthal averages are written out.Since the shear time t shear 5 0.2t eddy, with the conven-

tional pulsing scheme, the mean-field quantities havestabilized subsequent to the pulse before they are output.In order to focus on the relaxation process, we per-formed an additional pulsing experiment with the pulsesoccurring 0.5t shear before each 0.5t eddy output time.Thus, the outputs occurred during rather than after thesymmetrization process.


Figure 21 shows the mean azimuthal vertical velocityon z-surfaces z 5 0.75, 0.5, and 0.25 at T 5 0.5–2.5.One nondimensional unit of w corresponds to 1.0 m s21.For T 5 0.5–1.0, most of the vertical velocities are oforder 0.01–0.02 cm s21. These vertical velocities are toosmall to account for the observed temperature increaseof the vortex by mean subsidence warming alone duringthis period. In 1.0t eddy, the expected temperature changeof the vortex core due to mean subsidence is

u0 2Du ø N wt ø 0.075 2 0.15 K.eddyg

The observed warming in the first t eddy is about 2 K.By T 5 1.0, however, the z 5 0.25 level shows sub-

sidence of nearly 0.1 cm s21. Vertical velocities at theother displayed levels remain small until T 5 2.0, whenwe observe mean subsidence at all levels within theradius of maximum winds with maximum vertical ve-locities in the range 0.1–0.4 cm s21. Subsidence of order0.1 cm s21 then persists at all levels in the inner corethrough T 5 2.5. Thus, we expect subsidence warmingto be more significant at later times.

6. Comparison with observations

We have examined the results of a published rawin-sonde composite study of tropical cloud cluster evolu-tion and cyclogenesis in the western North Pacific (Lee1986, 1989a,b) In his study, Lee compared a compositeof 341 selected developing cyclones to 332 ‘‘nonper-sistent nongenesis’’ and 328 ‘‘persistent nongenesis’’tropical cloud clusters. Clusters that could be observedonly within one 24-h time period were defined as non-persistent nongenesis clusters, and those that were vis-ible for two or more days were termed persistent non-genesis clusters. To study time evolution, compositesfor the developing cyclones were formed for variousevolutionary time stages. The cyclone system was se-lected for inclusion in the composites when it was iden-tified as a tropical depression. This was defined as evo-lutionary stage 3. Stage 3 included the first two 12-htime periods following identification as a tropical de-pression. Stage 4 included the two 12-h periods im-mediately after stage 3, stage 2 included the 24 h im-mediately before stage 3, and stage 1 included the 24h prior to stage 2.

As a test of the relevance of our quasigeostrophicmodel, it is of interest to note the Rossby numbers atthe radius of maximum winds corresponding to Lee’sFig. 16, which shows a cross section of the azimuthalmean-tangential wind for the various stages of genesis.At genesis stage 1,

214 m sRo 5 5 0.26 K 1;

25 21 5(2.8 3 10 s )(5.5 3 10 m)

for this stage of genesis the quasigeostrophic approxi-mation is a good one. For genesis stage 3,

219 m sRo 5 5 1.3;

25 21 5(3.5 3 10 s )(2 3 10 m)

at this stage the quasigeostrophic approximation is nolonger strictly applicable.

The source of the increase in cyclonic circulation wasinvestigated using a tangential-momentum budget anal-ysis in a coordinate system moving with the storm sys-tem. Denoting the azimuthal mean-tangential wind byy , its depth-integrated time rate of change is given by

]ydp 5 mean terms 1 eddy termsE ]ttroposphere

1 motion terms 1 surface friction.

The mean, motion, and surface friction terms can bedetermined from the composited data (see Lee’s papersfor details). The eddy term is calculated as the residualof the other terms; it also includes any other residualeffects such as data errors. It is important to note thatany vertical fluxes of angular momentum are also in-cluded in Lee’s eddy term.

Lee’s tangential-momentum budget results are shownin Table 5. It is evident that for the genesis case thecontributions from the eddy terms are quite large com-pared to all other terms, particularly at stages 1 and 2.The importance of the eddy terms in the early stages ofcyclogenesis is consistent with our theory, which positslarge inward fluxes of eddy angular momentum on sub-synoptic scales (18–58 lat) subsequent to each convectiveburst. However, because Lee’s eddy term is determinedby the residual method, Lee’s large eddy fluxes cannotbe directly identified with our horizontal eddy fluxes;for example, vertical eddy fluxes could also contributeto Lee’s residual. The importance of eddy-momentumfluxes on the larger scale (58–98 lat) is not inconsistentwith suggested environmental influences on tropical cy-clogenesis (Challa and Pfeffer 1980, 1990; Pfeffer andChalla 1981; Montgomery and Farrell 1993).

7. Summary and conclusions

In this work we have examined the dynamics of theinteraction of moist penetrative convection with a larger-scale vortex. Convection is parameterized by its esti-mated net effect on the potential vorticity field. Con-vection could be initiated by environmental asymmetriessuch as those described by Challa and Pfeffer (1980,1990), Pfeffer and Challa (1981), Montgomery and Far-rell (1993), and Challa et al. (1998), or by mesoscaleprocesses. Here we provide an explicit explanation ofhow asymmetric convective-scale rotational kinetic en-ergy is transformed to kinetic energy of the large-scalevortex.

For simplicity, a three-dimensional quasigeostrophicbalance model is used to elucidate the underlying dy-namics. This approach has the virtue of excising gravitywaves, allowing one to focus exclusively on the rota-


FIG. 21. Azimuthal-mean vertical velocity w at model levels z 5 0.75, 0.5, and 0.25 at T 5 0.5, 1.0, 1.5,2.0, and 2.5 for a pulsed two-cluster experiment with pulses applied 0.5t shear before each output time. Onenondimensional vertical velocity unit equals 1 m s21. One nondimensional unit of radial displacement equals1000 km.


TABLE 5. Lee’s tangential momentum budget analysis (from Lee 1989b). Units are m s21 d21.

] u /]ty(m s21 d21) 5 Mean terms 1 Eddy terms 1 Motion terms 1 Surface friction

18–58Non-nonPer-nonGen-S1Gen-S2Gen-S3

00

11.712.011.8

55555

10.410.410.711.412.6

20.120.1

11.6 (70%)11.6 (53%)10.7 (21%)

20.220.220.320.420.4

20.120.120.320.621.1

58–98Non-nonPer-nonGen-S1Gen-S2Gen-S3

001.20.00.2

55555

20.110.110.410.310.4

10.610.4

12.0 (83%)11.2 (80%)11.4 (78%)

20.420.420.820.921.0

20.120.120.420.620.6

tional advective dynamics. Vortex axisymmetrizationproceeds by the ingestion of like-sign PV anomaliesaccompanied by the formation of PV filaments sur-rounding the larger-scale vortex as well as expulsion ofopposite-sign PV anomalies. Our results on merger oflike-sign PV anomalies are qualitatively consistent withtheoretical work on vortex merger in two-dimensionalEuler flow (Melander et al. 1988; Ritchie and Holland1993) and two-layer quasigeostrophic flow (Polvani1991). The dynamics of vortex Rossby waves are seento usefully characterize the axisymmetrization process,in that the vortex Rossby wave–mean flow theoreticalpredictions for the spinup magnitude and the locationof wave–mean flow acceleration are in good agreementwith the model results. Examination of Lagrangian tra-jectories, the Lorenz energy cycle, and Eliassen–Palmflux diagrams give a consistent picture of the processof axisymmetrization and vortex intensification.

With reasonable assumptions about the magnitude ofPV injection associated with moist penetrative convec-tion, we obtain spinup to a 15 m s21 cyclone on realistictimescales. Simulation of a midlevel vortex with pe-ripheral penetrative convection shows that axisymme-trization results in the spinup of a surface cyclone. Awarm-core vortex of order 5 K forms as a natural con-sequence of the axisymmetrization process. In the re-laxation experiments, both eddy-heat andeddy-momentum fluxes are found to contribute to thedevelopment of the warm core. In the pulsed convectiveexperiments, subsidence warming is initially too smallto account for the observed warming; however, after thefirst t eddy subsidence warming is significant.

At the finite amplitudes determined consistent withour calculation of the magnitude of PV injection, thespinup obtained shows greater than linear dependenceon the amplitude. This fact implies the existence of anonlinear feedback mechanism associated with convec-tively forced vortex Rossby waves. Discrete neutral orweakly unstable vortex Rossby modes are observed topropagate azimuthally around the vortex center both atupper and lower levels; the possible role that thesemodes may play in orchestrating future convection,

which may further intensify the vortex requires addi-tional study.

Numerical simulations of tropical cyclogenesis usinga full physics model (taking into account horizontal ad-vection, vertical advection, stretching, tilting of hori-zontal vorticity, and friction) show that the dominantcontributions to the low-level vertical vorticity tendencyare horizontal advection and stretching (Kurihara andTuleya 1981). This provides evidence that our idealizedmodel is not too idealized, that is, that we have incor-porated the fundamental processes producing thestorm’s vertical vorticity even though we have neglect-ed, for example, boundary layer friction and the com-plex mesoscale interactions leading to tilting of hori-zontal vorticity. Observations (Lee 1986, 1989a,b) sug-gest that small-scale eddy processes are important at theearly stages of tropical cyclogenesis. These results areconsistent with our expectations for vortex developmentby axisymmetrization in the presence of moist penetra-tive convection.

Further observations and carefully designed primitiveequation modeling experiments are needed, however, toconfirm or falsify the theory described here. These in-clude observations of patches of low-level cyclonic rel-ative vorticity in association with convection near theincipient vortex, such as a mesoscale convectively gen-erated vortex. While one of the ultimate goals of anobservational field program for tropical cyclogenesisshould be to construct finescale PV maps, for conve-nience we suggest the use of the absolute vorticity asa useful proxy for PV. A key process in our theory isthe shearing of such vorticity patches by the incipientvortex. Accompanying the shearing one would like toobserve the upgradient transport of vorticity into thevortex and the downgradient transport into surroundingvorticity filaments. As the dynamics of vortex Rossbywaves have been shown to usefully characterize thewave–mean flow interaction for the PV anomaly am-plitudes considered here, it would also be of interest toobserve wavelike features in the PV (or vorticity) fieldthat are consistent with their local dispersion relationderived in Montgomery and Kallenbach (1997). In prin-


ciple, this dispersion relation can be used to distinguishvortex Rossby waves from gravity–inertia waves in bothobservational and modeling data.

Future work will test the theoretical conclusions de-veloped here in a fully nonlinear three-dimensionalprimitive equation model and an asymmetric balancemodel, with moist processes represented in a more re-alistic manner.

Acknowledgments. We gratefully acknowledge Dr.Richard Johnson for his assistance in estimating tropicalconvective heating rates. We also acknowledge Dr.James McWilliams, Dr. Michael McIntyre, Dr. RaymondPierrehumbert, Dr. John Molinari, Dr. DominiqueMoeller, Dr. Vic Ooyama, Dr. Wayne Schubert, and Dr.Lloyd Shapiro for insightful comments and discussionsthat helped clarify the presentation. We thank Mr. PaulReasor for assistance in implementation of the nonlinearsemispectral model. We appreciate the computing sup-port provided by Mr. David Stepaniak and Mr. Rick Taft.This work was supported in part by National ScienceFoundation Grants ATM-9312655 and ATM-9529295.

APPENDIX A

The Quasigeostrophic Model

The numerical solution of Eqs. (4)–(6) follows thefinite difference method used by Montgomery and Far-rell (1992) suitably modified to simulate quasigeo-strophic dynamics. Time stepping is performed by atwo-step predictor–corrector Adams–Bashforth method(Gazdag 1976). The advective terms of Eqs. (4) and (6)are calculated using fourth-order Arakawa advection(Arakawa 1966) with the sign correction as noted byOrszag (1971) in order to minimize dispersion errors.All other derivatives employ second-order centered dif-ferences.

The vertical grid spacing dz 5 1.25 km, correspond-ing to nine vertical levels. The horizontal grid spacing(dx, dy) is typically 13.9 km, corresponding to about 14grid points inside the radius of maximum winds of theinitial circular basic-state vortex, and a total of 144 3144 total points in x and y. The Courant–Friedrichs–Lewy (CFL) condition then requires a time step lessthan or equal to 2.78 3 103 s. We generally used a timestep with a factor of 4–8 smaller than the CFL limit.For the single cluster relaxation (section 4c) and mid-level vortex (section 4d) configurations we decreasedthe horizontal grid spacing to 7.5 km by increasing thenumber of (x, y, z) points in the model to 200 3 2003 9 and shrinking the domain size from 2000 3 20003 10 to 1500 3 1500 3 10 km.

The model also includes second-order horizontal dif-fusion of Q. Diffusion is added to remove small-scalePV associated with the potential enstrophy cascade. Thevalue of the diffusion coefficient is based on two char-acteristic inviscid timescales, t eddy and t shear. The eddy1

2

turnover time of our barotropic vortex is initially 2.5 3105 s or approximately 2.91 days. The characteristicshear time is calculated based on the inverse of the localradial shear in a circular vortex,

21dV

t 5 r , (A.1)shear 1 2dr

where V 5 y (r)/r is the angular velocity of the circularvortex in geostrophic and hydrostatic balance. For ourvortex the maximum radial shear occurs initially neara nondimensional radius r 5 0.15 and has a dimensionalvalue of 0.18 3 1024 s21, resulting in a dimensionalshear time t shear 5 5.6 3 104 s. This yields a diffusivitybased on the vortex shear of nshear 5 (horizontal gridspacing)2/t shear 5 3.5 3 103 m2 s21. Corresponding tot eddy, with t eddy having a dimensional value of 2.5 31

2

105 s, we find neddy 5 (horizontal grid spacing)2/( t eddy)12

5 1.5 3 103 m2 s21. Based on these calculations andexamination of the model results at small scales, wechose n 5 4.2 3 102 m2 s21, a factor of 6 smaller thanthe mean value of neddy and nshear. The results presentedhere have been verified to be insensitive to the precisevalue of n.

In the absence of heating and friction the sum of thetotal kinetic energy K per unit mass plus the availablepotential energy P per unit mass integrated over thedomain is conserved. The potential enstrophy is alsoconserved if the vertical boundaries are isothermal, asis the case for all initial conditions used here. We expectto see a slight decrease in energy and potential enstrophywith time due to the explicit diffusion of PV in thenumerical model. Each of these quantities decreased byless than 0.5% in a 1.5 eddy-turnover-time run.

We tested our results for sensitivity to the time stepsize and the required convergence tolerance of the in-vertibility solver. No sensitivity to these parameters wasfound.

APPENDIX B

The Nondivergent Vorticity Equation Model

Although the nondivergent vorticity equation can besimulated with the quasigeostrophic model by simpli-fying to barotropic dynamics and interpreting f as thestreamfunction, the relative contribution from the wave–mean flow and wave–wave interaction are assessed di-rectly and naturally in the semispectral representation.In addition, comparatively higher temporal and spatialresolution can be obtained. The model employed ex-tends the linear numerical model of Montgomery andKallenbach (1997) to include the nonlinear advectiveterms. In the nondivergent model, the perturbationstreamfunction c9 and the perturbation relative vorticityz9 are represented semispectrally:


N

illc9(r, l, t) 5 c (r, t)e (B.1)O ll52N

andN

imlz9(r, l, t) 5 z (r, t)e , (B.2)O mm52N

where cl and zm denote the azimuthal Fourier amplitudefor streamfunction and vorticity, respectively, and anazimuthal wavenumber truncation of N 5 8 is used forthe example considered. The dimensional prognosticequation for the streamfunction is

]cn 22 ˆ5 ¹ F (r, t), (B.3)n n]t

where

1 ] ]c kF (r, t) 5 ik (c z ) 2 inz On k n2k n2k[ ]r ]r ]r|k|#N

|n2k|#N

dz dc

21 in c 2 in z 1 n¹ z , (B.4)n n n ndr dr

and n ± k and 5 (1/r)]/]r 1 ]2/]r2 2 n2/r2. A2¹n

diffusion coefficient n of 20 m2 s21 is used for thissimulation. The inversion is carried out using a standardtri-diagonal solver. Radial derivatives are computedwith centered second-order differences. Time steppingis performed using a fourth-order Runge–Kutta scheme(Abramowitz and Stegun 1972). The radial grid spacingis 8.5 km, with 200 radial points. Plots and results givenin this paper have been nondimensionalized as for thequasigeostrophic model.

APPENDIX C

Lagrangian Trajectories

To further elucidate the lateral mixing processes tak-ing place within the vortex, we track Lagrangian tra-jectories of PV particles, small elements of fluid thathave a fixed value of PV and are advected by the geo-strophic wind.2 The forward tracking algorithm findsthe end location of a particle with a given initial locationat t 5 0 after a given amount of time (typically themodel run time) has passed. The backward tracking al-gorithm finds, for a particle at a given location after themodel has run, the location from which the particleoriginated at t 5 0.

2 Strictly speaking, with the diffusion term included in the nu-merical integration of (4) Qg is no longer materially conserved. How-ever, for the examples considered, diffusion makes little differencesince setting diffusion coefficient to zero in the double-cluster ex-periment yields virtually identical trajectories.

To show how the trajectory algorithms are imple-mented, we first discuss forward tracking. We considerthe inital-value problem

dxi 5 u(x , y , t) (C.1)i idt

dyi 5 y(x , y , t) (C.2)i idt

for the ith PV particle, where u and y are the x and yvelocity fields calculated by the model, and xi(0) 5 aand yi(0) 5 b are given. Thus, for each of the i PVparticles we have a pair of coupled differential equa-tions, which we wish to solve for xi(t) and yi(t).

The equations are solved by fourth-order Runge–Kut-ta integration. To calculate the right-hand sides of Eqs.(C1) and (C2), gridded velocity fields at a chosen z levelare output from the quasigeostrophic model at each timestep. The values of u(xi, yi, t) and y(xi, yi, t) are obtainedfrom the gridded velocities by bilinear interpolation.

Backward trajectories are only slightly more com-plicated. The Runge–Kutta algorithm must step throughthe velocity fields from the end (typically, t 5 2 t E) of1

2

the file to its beginning at t 5 0. Because the trajectoriesare time-reversed (analogous to running a movie back-ward) the velocity vectors must also be reversed.

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