Meteorol Atmos Phys 97, 93–116 (2007)DOI 10.1007/s00703-006-0246-zPrinted in The Netherlands

International Pacific Research Center and Department of Meteorology, School of Oceanand Earth Science and Technology, University of Hawaii at Manoa, Honolulu, HI, USA

A multiply nested, movable mesh, fully compressible,nonhydrostatic tropical cyclone model – TCM4:Model description and development of asymmetrieswithout explicit asymmetric forcing

Y. Wang

With 16 Figures

Received August 21, 2006; accepted September 22, 2006Published online: March 14, 2007 # Springer-Verlag 2007

Summary

A newly developed, multiply nested, movable mesh, fullycompressible, nonhydrostatic tropical cyclone model –TCM4 is documented and used to investigate how theasymmetric structure develops in an initially axisymmetrictropical cyclone without any explicit asymmetric forcing.For this purpose, the model is configured on an f-plane andinitialized with an axisymmetric cyclonic vortex in a quies-cent environment with a constant sea surface temperature of29 �C. To isolate the effect of moist processes, a dry experi-ment with model physics turned off and a full physicsexperiment with full model physics included are performed.

The results show that the initial development of asymmet-ries results from the numerical finite-differencing schemeon a regular square grid system and time-splitting error as-sociated with the horizontal advection. The computationalasymmetries are dominated by azimuthal wavenumber-twoand wavenumber-four components. In the dry experiment,because the initial cyclonic vortex has a monotonic radialdistribution of potential vorticity and thus is dynamicallystable, the computational asymmetries could not grow fur-ther after their initiation and remain very weak and station-ary, thus having little effect on the overall evolution of thedry vortex in the model.

In the moist experiment, the initial development of asym-metries is quite similar to that in the dry experiment beforethe development of significant convection. However, onceconvection bursts near the original radius of maximum wind,the asymmetries are dominated by small-scale convective

activities near the original radius of maximum wind. Therapid convective outbursts generate strong gravity wavesthat propagate radially outward. After some dynamical ad-justments between 6 and 12 h of simulation, convectionis mainly trapped in the inner core region of the vortex.Convective heating associated with the inner core convectionproduces an off-center local PV maximum of the azimuthalmean vortex in the mid-lower troposphere. The reverse of theradial PV gradient across this PV annulus satisfies the neces-sary condition for barotropic instability, resulting in a rapiddevelopment of large amplitude asymmetries in the innercore region with low azimuthal wavenumbers. These asym-metries are characterized by vortex Rossby waves. It isalso found that these physical modes are slightly modifiedby quasi–stationary computational asymmetries. Althoughthe amplitude of the computational modes is relatively smallcompared to the physical modes, interpretation of the asym-metries in the inner core region at any given time, especiallythe wavenumber-two component, needs to be with caution.We have found no indication of contribution by inertialinstability in the outflow layer to the development of asym-metries in the inner core region of our simulated storm.

1. Introduction

The basic dynamics of the tropical cyclone can beunderstood based on the assumption of axisym-metric structure, comprising the highly rotational

tangential flow and the transverse secondary cir-culation. This is why many distinct features oftropical cyclones can be studied using axisym-metric models. In particular, the tropical cyclonecore is considered inertially very stable due tothe rapidly rotational flow which has strongshear in radial direction, therefore exerts strongdamping to any asymmetric forcing (Smith andMontgomery, 1995). As a result, to the first order,tropical cyclone can be considered as a quasi–axisymmetric, warm-cored cyclonic vortex. Thisis one of the basic assumptions used to developthe theoretical maximum potential intensity thata tropical cyclone can achieve (Emanuel, 1986,1988, 1995; Holland, 1997).

In reality, however, tropical cyclones are rarelyaxisymmetric. The asymmetric structure is gener-ally characterized by quasi–stationary or movingspiral rainbands (Willoughby et al, 1984), vortexRossby waves (Montgomery and Kallenbach,1997; Chen and Yau, 2001; Wang, 2001, 2002a,b), polygonal eyewalls (Lewis and Hawkins,1982; Schubert et al, 1999; Kuo et al, 1999), andmeso-vortices embedded in the eyewall (Blackand Marks, 1991; Willoughby and Black, 1996;Kossin et al, 2002; Montgomery et al, 2002).Although generally weak in the inner core of avery strong tropical cyclone, the asymmetries canbe considerably large in some situations, suchas when a tropical cyclone is embedded in ver-tical shear or in a strong steering flow (Wangand Holland, 1996a; Bender, 1997; Frank andRitchie, 1999, 2001).

Although an order in magnitude smaller thanthe primary circulation, the asymmetries in theinner core are dynamically important in modi-fying both the structure and intensity of the tro-pical cyclone (see a recent review by Wangand Wu (2004)). Montgomery and Kallenbach(1997) proposed that axisymmetrization of asym-metries in the inner-core region could lead atropical cyclone to intensify via vortex Rossbywave-mean flow interaction. In a three-dimen-sional quasi–geostrophic model, Montgomery andEnagonio (1998) showed that a tropical-cyclonevortex could intensify when convectively gener-ated potential vorticity (PV) anomalies of likesign were ingested into the parent vortex core,suggesting that asymmetries contribute to the tro-pical cyclone intensification in the early develop-ment stage. The positive role of the inner-core

asymmetric eddies in the tropical cyclone inten-sification has also been demonstrated from ob-servations. Heymsfield et al (2001) analyzed theasymmetric structure of the intense convection inhurricane Bonnie (1998) based on multiple obser-vations synthesized from the National Aeronau-tics and Space Administration (NASA) ER-2and DC-8 aircraft. They found that significantinner-core asymmetries occurred just before rapidintensification.

In contrary, Schubert et al (1999), and Nolanand Montgomery (2002) suggested that directspin-down of the eyewall by eddies caused bybarotropic vortex breakdown could weaken the in-tensity of a well-defined tropical cyclone. Usinga limited-area primitive equation model, Penget al (1999) studied the effect of the planetaryvorticity gradient and an imposed uniform meanbackground flow on the intensity of a tropicalcyclone. They found that the most intense tropi-cal cyclone evolved on an f-plane in a quiescentenvironment and inclusion of the beta-effect ora uniform background flow induced a wavenum-ber-one asymmetric structure in the inner coreand reduced the tropical cyclone intensity. Witha hydrostatic primitive equation model, Wu andBraun (2004) investigated the influence of a uni-form environmental flow, the beta effect, and ver-tical shear of the environmental flow on tropicalcyclone maximum intensity. They showed thatthe eddy momentum flux associated with inner-core asymmetries could weaken a TC directlyby producing a deceleration of the azimuthalmean tangential wind in the vicinity of radiusof maximum wind and indirectly by inducingan anomalous secondary circulation opposite ofthe primary secondary circulation. Recently Yanget al (2007) find that even in the absence ofany environmental forcing, the simulated tropicalcyclone maximum intensity could be about of15% weaker in a three-dimensional tropical cy-clone model than in its axisymmetric version.This maximum intensity reduction is attributedto the existence of asymmetric eddies charac-terized by vortex Rossby waves in the inner corein the three-dimensional simulation while theirabsence in the axisymmetric simulation. Thesestudies thus suggest that the inner-core asymmet-ries could be a dynamical limiting factor to thetropical cyclone maximum intensity while couldpositively contribute to the rapid tropical cyclone

94 Y. Wang

intensification in the early development stage.Therefore, understanding the formation mechan-ism of the inner-core asymmetric structure isimportant to study the tropical cyclone structureand intensity changes.

Previous studies have shown that the asym-metric structure in the inner core of a tropical cy-clone can be generated by external forcing, suchas the beta effect (Wang and Holland, 1996b, c;Bender, 1997; Peng et al, 1999) and large-scaleenvironmental flow (Shapiro, 1983) and its ver-tical shear (Wang and Holland, 1996a; Bender,1997; Frank and Ritchie, 1999, 2001), or internaldynamics (e.g., Schubert et al, 1999). Most ofrecent studies using full physics models havefocused mainly on the generation of inner-coreasymmetric structure due to external forcing,such as the large-scale environmental flow or itsvertical shear (Wang and Holland, 1996a; Bender,1997; Peng et al, 1999; Frank and Ritchie, 1999,2001). Recent studies show that the inner-coreasymmetries can develop even in the absenceof any external forcing in full physics models(Chen and Yau, 2001; Wang, 2001, 2002a, b;Yang et al, 2007) and is often attributed to baro-tropic instability of the mean vortex (Schubertet al, 1999). A detailed analysis of the initiationand development of the inner-core asymmetriesin a full physics model, however, has been lack-ing. The questions are occasionally asked whetherthese asymmetries are physically or numericallygenerated in these models. The objective of thisstudy is to try to answer this question and pro-vide a detailed analysis of the development ofthe inner-core asymmetric structure in a nu-merically simulated tropical cyclone in a newhigh-resolution, nonhydrostatic tropical cyclonemodel (TCM4).

In the absence of any external asymmetricforcing, the dynamical instability is the onlysource that can lead to any infinitely small asym-metric perturbations to grow in a symmetrictropical cyclone vortex. In a numerical model,the initial infinite-amplitude perturbations canbe induced by numerical truncation errors, orlateral boundary conditions in a limited areamodel. If the basic symmetric vortex is dynami-cally stable, these errors could not grow and arethus expected to be negligibly small. However,if the symmetric vortex is dynamically unstable,these infinite-amplitude perturbations may grow

depending on their azimuthal wavenumbers.Those with azimuthal wavenumbers close tothat of the most unstable modes will grow muchfaster and becomes the dominant asymmetricmodes. Previous studies have identified twosuch dynamical instabilities, namely the inertialinstability in the outflow layer and barotropicinstability in the mid-lower troposphere.

In the early days of numerical modeling stud-ies using three-dimensional models with rela-tively coarse horizontal and vertical resolutions,the development of asymmetries were mainly at-tributed to the inertial instability in the outflowlayer (Anthes et al, 1971; Anthes, 1972). Suchinertial instability was found after the modeltropical cyclone intensified strong enough to de-velop a strong outflow jet with regions of nega-tive absolute vorticity. The release of the inertialinstability was considered to contribute to theformation of outer spiral rainbands and the an-ticyclonic looping motion of the model tropi-cal cyclone (Anthes et al, 1971; Anthes, 1972).Anthes (1972) showed that the asymmetric eddiesdeveloped in the outflow layer derived theirkinetic energy from the mean symmetric flow.Kurihara and Tuleya (1974) revisited the eddydynamics in their three-dimensional tropicalcyclone model. They found that the horizontalasymmetric eddies occurred predominantly in theoutflow layer and that eddies which developed intheir model were maintained in the mature stageby energy supply from both mean kinetic energyand total potential energy, while the supply frompotential energy was more important duringthe early development stage of the storm. Theseearlier studies focused mainly on the large-scaleasymmetries in the outflow layer, which werebelieved to originate from the inertial instability.

The tropical cyclone formation and mainte-nance rely on the energy transferred from theocean surface and released as latent heat in moistconvection in the eyewall only tens of kilometersfrom its centre. In the lower troposphere, air par-cels flowing inward and spiraling upward in theconvective eyewall usually experience a materialincrease in PV (Schubert et al, 1999). Since thismaterial increase of PV can be especially rapidnear the eyewall, an annular tower of high PV

with low PV in the central eye region could bepossible if there are no asymmetries in eyewallconvection at the beginning, and if the material

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 95

PV concentration could continue without anyother processes to mix the PV between the eyeand the eyewall. Thus, barotropic instability nearthe eyewall can be expected. This type of in-stability and its potential role in causing eyewallbreakdown and polygonal eyewall structure havebeen investigated by Schubert et al (1999) with anondivergent barotropic model. They found thatwhen the instabilities grow to finite amplitude,the vorticity in the eyewall pools into discreteareas, creating the appearance of polygonal eye-walls. Schubert et al, suggested that the circula-tions associated with these vorticity pools exhibita connection to tropical cyclone mesovorticesin the eyewall, as observed in some real hurri-canes. At later times, the vorticity is redistributedinto a nearly monopolar circular vortex througheye contraction and PV mixing since there is noPV source in their dry nondivergent barotropicmodel.

In addition to the dynamical instabilities,Tripoli (1992) noticed that in his hurricane model,the asymmetries were initiated in the inner coreby the finite-differencing truncation error with adistinct azimuthal wavenumber-4 perturbation dueto the rectangular grid geometry. He found thatthis small numerical truncation error grew rapidlyto a distinct pattern of four dominant convectivecenters appeared by 10 h of simulation at the fin-est grid spacing of 10 km. Since he activated the3.3 km grid simulation after 8 h of integration, it isnot clear whether such numerical truncation errorscould be reduced if the initial simulation startedfrom the finest grid spacing.

The objectives of this study are to documentthe newly developed, multiply nested, movablemesh, fully compressible, nonhydrostatic tropicalcyclone model (TCM4) and to use it to examinethe details of both initiation and development ofasymmetries in the inner core of a simulated tro-pical cyclone in the absence of any physicallyasymmetric forcing. The rest of the paper is orga-nized as follows. Section 2 describes the tropicalcyclone model that the author has recently de-veloped and the design of the numerical experi-ments. Sections 3 and 4 discuss the initiationand development of asymmetries in the innercore of an initially axisymmetric tropical cyclonevortex in dry and moist experiments, respec-tively. The major findings are summarized in thelast section.

2. Model description and experimental design

2.1 Model description

The new tropical cyclone model (TCM4) usesthe fully compressible, water-loading, nonhydro-static equations formulated in Cartesian coordi-nates in the horizontal and the mass coordinate inthe vertical. The governing equations in z-coor-dinate can be written as

du

dtþ Cp�M

@�

@x¼ f v� ~ff wþ Du þ Fu

dv

dtþ Cp�M

@�

@y¼ �fuþ Dv þ Fv

dw

dtþ Cp�M

@�

@z¼ �gþ ~ff uþ Dw þ Fw;

d�

dt¼ Q

Cp�þ D� þ F�

d�

dtþ R�

Cv

r3 � ~VV ¼ R�

Cv�v

d�vdt

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

ð1Þ

where u, �, w are zonal, meridional, and verti-cal wind components, respectively; x, y, z, andt are three-dimensional coordinates and time;f (¼ 2� sin’) and ~ff (¼ 2� cos’) are verti-cal and horizontal Coriolis parameters; �, �v[¼ �ð1 þ 0:608qvÞ] are potential temperature andvirtual potential temperature, respectively; � isthe Exner function; R is the gas constant of dryair; Cp, Cv are the specific heat at constant pres-sure and volume, respectively; Du, Dv, Dw, D�

are subgrid horizontal diffusion of zonal, meri-dional, vertical winds and potential temperature;Fu, Fv, Fw, and F� are subgrid vertical mixing ofzonal, meridional and vertical winds and poten-tial temperature; r3 � ~VV is the three-dimensionaldivergence of three-dimensional winds; �M is theliquid potential temperature defined as

�M ¼ �Qv ¼ �1 þ 1:608qv

1 þ qv þ qc þ qr þ qi þ qs þ qg;

ð2Þwhere qv, qc, qr, qi, qs, and qg are mixing ratiosof water vapor, cloud water, rain water, cloud ice,snow, and graupel, respectively.

To facilitate the time integration, we separateeach prognostic variable into a motionless basicstate and the deviation from the basic state,

� ¼ �þ �0

� ¼ �þ �0

�; ð3Þ

96 Y. Wang

where � ¼ ½ �ppðzÞ=p00�R=Cp and �ppðzÞ is the unper-turbed pressure.

Also we introduce the mass vertical coordinate� defined as � ¼ �pp=�pps, with �pps being the unper-turbed surface pressure which can be a functionof x and y due to the presence of any terrain. Inthe mass vertical coordinate, the governing equa-tions can be rewritten as

du

dtþCp�M

�@�0

@x� @ lnps

@x

@�0

@ ln�

�

¼ f v� ~ff wþDu þ Fu

dv

dtþCp�M

�@�0

@y� @ lnps

@y

@�0

@ ln�

�

¼ �fuþDv þ Fv

dw

dt� g�Cp�

R��

@�0

@�¼ g

��M�

� 1

�

�~ff uþDw þ Fw;d�0

dtþ _��

@�

@�¼ Q

Cp�þD� þ F�

d�0

dt� g��

Cv��

@w

@�� gw

Cp�

¼ �R�

Cv

r �~VV þ R�

Cv�

�Q

Cp�þD� þ F�

�

8>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð4Þ

where

_�� ¼ � g�w

R��� �

�u@ ln ps

@xþ v

@ ln ps

@y

�; ð5Þ

r �~VV ¼ @u

@xþ @v

@y��@ lnps

@x

@u

@ ln�þ@ lnps

@y

@v

@ ln�

�;

ð6Þ

d

dt¼ @

@tþ u

@

@xþ v

@

@yþ _��

@

@�: ð7Þ

The time-splitting method originally introducedin a three-time-level time integration scheme byKlemp and Wilhelmson (1978) and later modi-fied for a two-time-level scheme by Wicker andSkamarock (1998; 2002) is used for the modeltime integration. By this time-splitting method,terms related to the advection and model phy-sics are integrated with large time step. Termsassociated with the horizontal propagating iner-tia-gravity waves are integrated using the for-ward-backward scheme and those associatedwith the vertically propagating sound waves areintegrated using the implicit scheme, both with

fractional small time steps. As a result, the gov-erning Eq. (4) can be rewritten as

@u

@tþCp�M

�@�0

@x� @ lnps

@x

@�0

@ ln�

�� f vþ~ff w

¼ AuþDuþFu

@v

@tþCp�M

�@�0

@y� @ lnps

@y

@�0

@ ln�

�þ fu

¼ AvþDv þFv

@w

@t� g�Cp�

R��

@�0

@�� g

��M�

� 1

��~ff u

¼ AwþDw þFw;@�0

@tþ _��

@�

@�¼ A�þ

Q

Cp�þD� þF�

@�0

@t� g��

Cv��

@w

@�� gw

Cp�þR�

Cv

r �~VV

¼ A�0 þR�

Cv�

�Q

Cp�þD�þF�

�

8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð8Þ

where Ax represents the horizontal and verticaladvection of variable x. All terms on the right-hand side are thus associated with the low-fre-quency modes and are evaluated with large timestep, while terms on the left-hand side are asso-ciated with high-frequency modes and are eval-uated at fractional small time steps. Note that inthe thermodynamic equation, the vertical advec-tion of the unperturbed potential temperature iscalculated at small time steps to allow the ther-modynamic adjustment related to the verticallypropagating sound waves. Our sensitivity experi-ments show that calculation of this term withlarge-time step may cause numerical instabilityand thus a constraint on model time step. Notealso that mixing ratios of all hydrometeors, tur-bulent kinetic energy and its dissipation rate areall integrated with large time step. The tendencyof Qv is calculated with large time step but the Qv

itself in (2) is updated during the fractional smalltime steps in order to get accurate liquid potentialtemperature �M.

The time tendencies due to horizontal ad-vection in (8) are calculated with the fifth-orderupstream advection scheme, an extension of theoriginal third-order scheme developed by Wang(1996). The fifth-order has much weaker numer-ical dissipation, smaller dispersion error, andbetter shape-conserving properties than both thecentered finite-difference scheme used in theprevious TCM3 and the third-order scheme

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 97

discussed in Wang (1996). Tendencies due tovertical advection are calculated with the sec-ond-order upstream scheme of Wang (1996). Tofacilitate the description of the whole time inte-gration scheme, we refer to �T as the large timestep, T as the current model time, and �t as thefractional small time step. We also assume N asthe number of small time steps for each largetime step, thus N ¼ �T=�t, while we assume nas the index of the current small time step. Withthese notations, the time integration of (8) can besummarized as follows:

(1) To calculate all the advective tendency terms

ðATþ�T=2x Þ without contributions from iner-

tia-gravity waves and sound waves.(2) To update potential temperature and liquid po-

tential temperature and get �Tþn�t and �Tþn�tM .

(3) Using �Tþn�t, �Tþn�tM , and �0

Tþðn�1Þ�tto get

uTþn�t and vTþn�t.(4) To calculate r � ~VVTþn�t.(5) To solve w and �0 equations implicitly to get

wTþn�t and �0Tþn�t using the updated �, �M,

and horizontal divergence r � ~VV , and an un-centered time averaging for w and �0 is de-fined as following:

�0Tþðnþ1=2Þ�t¼1

2ð1þ�Þ�0Tþn�tþ1

2��0Tþðn�1Þ�t;

wTþðnþ1=2Þ�t¼1

2ð1þ�ÞwTþn�tþ1

2�wTþðn�1Þ�t:

ð9Þ(6) To subtract the changes due to advection

from the updated new fields and to recalcu-

late the advective tendencies at n¼N=2.These new advective tendencies include con-tributions from the inertia-gravity waves andsound waves.

(7) To use the new advective tendencies to re-peat steps (2)–(5) until the n¼N to completeone large time step ð�TÞ integration.

The number of small time steps within eachlarge time step must be even (in our application,we set N¼ 8). A similar Crowley time-splittingscheme has been described by Wicker andSkamarock (2002). Note that the horizontal Cor-iolis force terms in (8) are calculated implicitlyas well in our implementation.

The model physics are the same as those usedin TCM3 (see Table 1 and Wang, 2001; 2002c).They include anE-" turbulence closure scheme forsubgrid scale vertical turbulent mixing; a modifiedMonin-Obukhov scheme for the surface flux cal-culation (Fairall et al, 2003); an explicit treatmentof bulk cloud microphysics package, which in-cludes mixed phase cloud processes (Wang, 1999;2001); a deformation-dependent fourth-order hor-izontal diffusion for all prognostic variables ex-cept for �0; and a simple Newtonian cooling termis added to the potential temperature equation tomimic the radiative cooling in the model as used inTCM3 and in Rotunno and Emanuel (1987). Notethat the model has a flat surface with an unper-turbed surface pressure of 1010 hPa. The modeltop is set at about 38 km and a sponge upperboundary condition similar to that used in Durranand Klemp (1983) is used to absorb the upwardpropagating sound and gravity waves.

Table 1. Summary of the model physics in TCM4

Physical process Scheme Reference Comments

Moist processes Bulk mixed-phasecloud microphysics

Wang (1999; 2001) Based mainly on Lin et al(1983), Rutledge and Hobbs(1983), and Reisner et al (1998)

Vertical mixing E-" turbulenceclosure scheme

Langland and Liou (1996) Modified to include cloudbuoyancy production ofturbulence (Wang, 1999)

Horizontal diffusion Deformation-dependentfourth-order

Wang (2001)

Surface layer over ocean Bulk scheme Fairall et al (2003) TOGA COARE v3.0Radiation Simple Newtonian cooling Rotunno and Emanuel (1987) Relaxation time is 12 hDissipative heating Turbulent kinetic energy

(TKE) dissipation rate " isadded to thermodynamicequation as dissipative heating

98 Y. Wang

The model domain can be multiply nested withtwo-way interactive nesting and with the innermeshes automatically moving following the mod-el tropical cyclone as used in TCM3 (Wang,2001). As in Wang (2001), the same model phy-sics are used in all meshes. Since no large-scaleenvironmental flow is included in this study, con-vection is mainly active in the inner core regionand in the spiral rainbands that are within abouta radius of 200 km from the cyclone center andthus can be covered in the finest innermost do-main. Therefore, cumulus parameterization is notconsidered even in the two outermost coarsemeshes in this study.

2.2 Experimental design

The experimental design follows Wang (2001;2002a). The model is initialized with an axisym-metric cyclonic vortex on an f-plane of 18� N ina quiescent environment. The initial thermody-namic structure of the unperturbed model atmo-sphere is defined as the western Pacific clear-skyenvironment given by Gray et al (1975). Thetangential wind of the initial cyclonic vortex(Wang and Li, 1993) is defined by

VTðr; �Þ ¼ VðrÞ sin�

2

�� �T

1 � �T

� �� �; �>�u;

0; ���u;

8<:

ð10Þwhere �u ¼ 0:15 and

VðrÞ ¼

Vm

�r

rm

��exp

1

b

�1 �

�r

rm

�b�

� jr � rmjRo � rm

exp1

b

�1 �

�Ro

rm

�b��;

r�Ro;0; r>Ro;

8>>>>>><>>>>>>:

ð11Þwhere Vm is the maximum tangential wind atthe radius rm, r the radius, b is a nondimensionalparameter which determines the rate of radialdecay of tangential wind outside the radius ofmaximum wind, and Ro the radius out of whichthe vortex wind vanishes. The mass and ther-modynamic fields associated with the vortexare obtained by solving the nonlinear balanceequation as described in the Appendix of Wang(2001).

The model domain covers a large area so thatthe effect of the lateral boundary conditions onthe interior is minimal. In our current model set-tings, the model domain is quadruply nested withresolutions of 67.5, 22.5, 7.5, 2.5 km for thefour meshes, respectively. The sizes of the fourmeshes are 201�181, 109�109, 127�127, 163�163 from the outermost mesh to the innermostmesh. The model has 26 � levels in the vertical(Table 2) with vertically staggered grid such thathorizontal winds, perturbation pressure and po-tential temperature and all moist variables arelocated at the integer levels while the verticalwind and turbulent kinetic energy and its dissipa-tion rate are arranged at the half levels.

Two experiments are conducted in this study.In the first experiment, the model is run in adry mode with all model physics turned off ex-cept that the horizontal diffusion is retained. Inthis dry experiment, we set Vm¼ 30 ms�1, rm¼40 km, b¼ 0.5, and Ro¼ 900 km to mimic atropical-cyclone-like vortex. In the other experi-ment, all model physics are retained, and thuswe call it the full physics experiment. In orderto simulate the intensification of the model stormin the full physics experiment, we set Vm¼20 ms�1, rm¼ 80 km, b¼ 1.0, and Ro¼ 900 km.In both experiments, the sea-surface temperatureis fixed at 29 �C and an f-plane is assumed andthe horizontal Coriolis forces are ignored so thatthere is no explicit asymmetric forcing to themodel storm. The dry experiment is integratedfor 72 h while the full physics experiment is inte-

Table 2. List of the model integer vertical levels

Level � Height(m)

Level � Height(m)

1 0.006 35396.43 14 0.450 6606.092 0.019 27270.02 15 0.510 5630.643 0.035 23081.21 16 0.570 4746.114 0.055 20119.25 17 0.630 3935.275 0.078 17962.95 18 0.690 3185.076 0.103 16340.12 19 0.750 2486.697 0.130 14994.82 20 0.806 1876.048 0.159 13791.27 21 0.856 1360.179 0.192 12614.18 22 0.900 926.33

10 0.230 11436.86 23 0.936 583.8911 0.275 10221.12 24 0.964 324.6712 0.330 8927.18 25 0.984 143.1313 0.390 7694.90 26 0.996 35.60

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 99

grated for 120 h to cover both the intensificationand mature stages of the model tropical cyclone.

Determination of tropical cyclone center iscritical in examining the inner core asymmetricstructure in a numerical model. We tested severaldifferent methods and found that the mass centercan provide very promising results at very high

resolutions. The mass center in the model is de-fined as the column mass centroid given by

X ¼ÐpsxdxdyÐpssdxdy

; Y ¼ÐpsydxdyÐpsdxdy

; ð12Þ

where ps is surface pressure. The domain ofintegration is a circular area whose radius is

Fig. 1. Total (top panels), wavenumber-two(middle panels), and wavenumber-four (bottompanels) asymmetries in pressure (left) andwinds (right) at 1 km height above the sea sur-face after 24 h of simulation in the dry experi-ment. Contour interval is 0.5 Pa in pressureperturbations. The wind vector is scaled in0.2 m s�1

100 Y. Wang

approximately 180 km from the surface vortexcenter. Since the integration domain is a func-tion of the vortex center itself, several iterationsare needed to reach a convergent solution of thevortex center by this approach. Given the firstguess of the vortex center from previous timestep, the solution converges very quickly with4–5 iterations.

3. Dry experiment

Because the initial cyclonic vortex has a mono-tonic radial distribution of PV, the model vortexis thus dynamically stable to any asymmetric per-turbation in the dry experiment. Since there is noany explicit asymmetric forcing on an f-plane ina quiescent environment for an initially axisym-metric vortex, any asymmetries that developed

Fig. 2. Total (top panels), wavenumber-two(middle panels), and wavenumber-four (bottompanels) asymmetries in potential vorticity (left)and vertical velocity (right) at 1 km heightabove the sea surface after 24 h of simulationin the dry experiment. Contour intervals are0.05 PVU (1 PVU¼ 10�6 K m2 kg s�1) in (a)and (b), and 0.01 PVU in (c) and 3 mm s�1 inasymmetric vertical motion

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 101

in the dry experiment must be of numerical notphysical origin. There are two such computationalsources: one is the use of the rectangular modeldomain with quadruply two-way interactive nest-ing, the other is the time-splitting error of horizon-tal advection in the zonal and meridional directions(see also Tripoli, 1992). These numerical asymmet-ries are dominated by even azimuthal wavenum-bers, such as wavenumbers two and four, etc. andtherefore the model vortex remains stationarythroughout the 72 h integration (not shown).

An example after 24-h integration is given inFigs. 1 and 2, which show the total asymmetricpressure and wind fields and their wavenumber-two and wavenumber-four components at 1 kmheight (Fig. 1) and the corresponding asymmetric

Fig. 3. Time evolution in the amplitudes of wavenumber-two, wavenumber-four, and the residual asymmetric pres-sure (Pa) in the dry experiment

Fig. 4. Height-time Hovm€ooller diagram for thetotal (a), wavenumber-two (b), and wavenum-ber-four (c) asymmetric kinetic energy averagedwithin a radius of 200 km in the dry experi-ment. Contour interval is 2�10�4 m2 s�2

102 Y. Wang

PV and vertical velocity fields at the same height(Fig. 2). The asymmetries are quasi–stationaryand have a structure very similar to that shownin Figs. 1 and 2 at any given time (not shown).Namely, positive asymmetric pressure anomalieslie in the northwest and southeast quadrants andnegative asymmetric pressure anomalies lie inthe northeast and southwest quadrants within theradius of maximum wind, which is 40 km fromthe cyclone center (section 2). The asymmetriesare dominated by wavenumber-two and wave-number-four structures in the azimuth with amaximum pressure perturbation of about 4 Pa,

about 15–20 km from the cyclone center. Notethat the asymmetric winds seem to be quasi–bal-anced with the asymmetric pressure fields. Thecyclonic (anticyclonic) circulation correspondsto negative (positive) asymmetric pressure per-turbation (Fig. 1a and d, and b and e). The wave-number four component exhibits a quite differentrelationship between the pressure and wind fieldswith convergence and divergence between thehigh and low pressure centers, much the sameas stationary gravity waves (Fig. 1f).

The asymmetric PV anomalies are negativelycorrelated with asymmetric pressure perturba-

Fig. 5. Radial-vertical cross-sections of total(top panels), wavenumber-two (middle panels),and wavenumber-four (bottom panels) asym-metric kinetic energy after 1 h (left) and 48 h(right) of simulations, respectively, in dry ex-periment. Contour intervals are 2�10�3 m2 s�2

in top and middle panels and 1�10�3 m2 s�2 inbottom panels

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tions (Figs. 1a and 2a, and 1b and 2b). This ishowever not the case for wavenumber four, whichshows a positive correlation between the pressureperturbations and the PV anomalies (Figs. 1c and2c) with much weaker anomalous PV than wave-number two (Fig. 2b and c). The wavenumberfour shows stronger upward and downward mo-tions in the inner core region than wavenumbertwo (Fig. 2e and f). The amplitude of the verticalvelocity, however, is quite weak and only about9 mm s�1 for wavenumber four and 3 mm s�1 forwavenumber two.

The asymmetries shown in Figs. 1 and 2 areinitiated from the first several time steps of themodel integration and remain very weak andquasi–stationary up to 72-h integration. This canbe seen from Fig. 3 which shows the evolutionof the amplitudes of pressure perturbations forwavenumbers two and four and the residual at100 m height. As indicated above, the asym-metric pressure perturbation is dominated bywavenumber-two component, followed by wave-number-four, with negligible residual.

The asymmetries develop two modes in thevertical after their initiation in lower levels. Asseen from the asymmetric (eddy) kinetic energyshown in Fig. 4, the asymmetries first develop inthe lower troposphere, which is followed by theslow development of asymmetries in the uppertroposphere with the maximum amplitude atabout 10–12 km height. This second maximumin eddy kinetic energy is also dominated by azi-muthal wavenumber-two component (Fig. 4b).The larger amplitude occurs in the lower tropo-sphere for wavenumber two (Fig. 4b) but in theupper troposphere for wavenumber four (Fig.4c). Note that the asymmetric structure seemsto reach a quasi–steady state after about 48-hintegration. Similar to those in the lower tropo-sphere (Fig. 5a–c), the asymmetric eddies in theupper troposphere also occur within the radiusof maximum wind, with their maximum eddykinetic energy at the radius of about 20 km fromthe cyclone center (Fig. 5d–f). A small outwardtilt of the asymmetric eddy kinetic energy is alsovisible (Fig. 5d–f).

The maximum eddy kinetic energy is only about0.01 m2 s�2, about five-order smaller than thekinetic energy of the mean vortex (450 m2 s�2).These computational asymmetries are thus tooweak to have any significant effect on the overall

evolution of the dry vortex in the model. Indeed,the model vortex remains its structure and inten-sity quite well in the dry experiment (not shown).The question arises as to whether these numericalasymmetries contribute to the initiation and de-velopment of asymmetries in the more realisticfull physics simulation. This will be addressed inthe following section.

4. Full physics experiment

Figure 6 shows the maximum wind speed (dashedcurve) at the lowest model level (35.6 m above thesea surface) and the minimum central sea-levelpressure (solid curve) of the simulated tropicalcyclone in the full physics experiment. The stormintensifies rapidly after an initial adjustment ofabout 6 h in response to the spin-up of the bound-ary layer and moist processes. The storm reachesits quasi–steady evolution after about 3-day sim-ulation but with some intensity oscillations after-wards, in particular in maximum surface wind.The storm reaches a maximum near surface windof about 74 m s�1 and a minimum central sealevel pressure of about 920 hPa in the 5-daysimulation. The axisymmetric structure of thestorm after 3-day simulation is shown in Fig. 7,including the tangential, radial winds, verticalvelocity, temperature anomalies, PV, and theequivalent potential temperature. The storm has

Fig. 6. Temporal evolution of the maximum wind speed inm s�1 (dashed curve and right legend) at the lowest modellevel (35.6 m above the sea surface) and the minimum centralsea-level pressure in hPa (solid curve and left legend) of thesimulated tropical cyclone in the full physics experiment

104 Y. Wang

its maximum tangential wind at a radius of about18 km (Fig. 7a), a shallow inflow layer in theboundary layer and a relatively deep outflow lay-er in the upper troposphere (Fig. 7b). The eye-

wall ascent tilts radially outward with height(Fig. 7c). The storm has a warm-cored structurein the mid-upper troposphere (Fig. 7d) and anoff-centered PV maximum just within the radius

Fig. 7. The azimuthal mean structure of the simulated tropical cyclone after 72 h of simulation in the full physics experiment:(a) tangential wind (contour interval 5 m s�1), (b) radial wind (contour interval 2.5 m s�1), (c) vertical velocity (contourinterval 0.5 m s�1), (d) perturbation temperature (contour interval 2 K), (e) potential vorticity (contour interval 20 PVU), and(f) equivalent potential temperature (contour interval 5 K)

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 105

Fig. 8. The wavenumber-one asymmetric surface pressure fields simulated in the full physics experiment with every hourinterval from 1 h to 20 h of simulation. Note that contour intervals are 0.2 Pa for t¼ 1 h to t¼ 6 h, 1 Pa for t¼ 7 h, 0.1 hPa fort¼ 8 to t¼ 13 h, and 0.2 hPa for t¼ 14 h to t¼ 20 h

106 Y. Wang

Fig. 9. The wavenumber-two asymmetric surface pressure fields simulated in the full physics experiment with every hourinterval from 1 h to 20 h of simulation. Note that contour intervals are 0.2 Pa for t¼ 1 h to t¼ 4 h, 1 Pa for t¼ 5 h, 5 Pa fort¼ 6 h, and 0.2 hPa for t¼ 7 h to t¼ 20 h

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 107

Fig. 10. Instant rainfall rate in mm h�1 at every hour interval from 4 h to 23 h of simulation in the full physics experiment

108 Y. Wang

of maximum wind (Fig. 7e). The equivalent po-tential temperature is quite uniform in the eye-wall with smaller values outside the eyewall anda relatively small pool in the eye region in themiddle troposphere (Fig. 7f). These features areconsistent with both observations and previousmodel results.

Similar to those in the dry experiment dis-cussed in the last section, asymmetries are alsodeveloped in the full physics experiment. How-ever, unlike in the dry experiment, both wave-number one and wavenumber two develop intothe dominant asymmetric components in the fullphysics experiment after about 12 h of simulation(Figs. 8 and 9). Before the initiation of any pre-cipitation during the first 3 h, the model vortexbehaves like a dry one and thus the asymmetriesdeveloped in the model are dominated by veryweak quasi–stationary wavenumber-two compo-nent (Fig. 9) with negligible wavenumber-onecomponent (Fig. 8). However, once the precip-itation started after about 4 h of simulation(Fig. 10), both wavenumber-one and wavenum-ber-two asymmetries (and higher wavenumberasymmetries as well) develop much complicatedstructures until 12 h of simulation. This is fol-lowed by well organized precipitation (Fig. 10)as well as asymmetric structure in the inner coreregion of the developing storm (Figs. 8 and 9).Note that the asymmetries are no longer quasi–stationary but propagate cyclonically along theeyewall of the storm.

As we see from Fig. 11, the asymmetries inthe full physics experiment are dominated bywavenumber-one and wavenumber-two compo-nents in pressure perturbations. The amplitudesof the asymmetries (Fig. 11), in particular thoseof wavenumber one and wavenumber two, growas the storm intensifies during the first 3 daysof simulation (Fig. 6). Examination of the radialPV structure of the azimuthal mean vortex showsthat as the storm intensifies, the vortex developslocal PV maximum near the inner edge of theeyewall (Fig. 12), indicating that the storm be-comes barotropically unstable and favors the de-velopment of asymmetric eddies in the eyewall(Schubert et al, 1999). As a result, the asym-metries, which are dominated by azimuthal lowwavenumbers, develop partly by subtracting theirkinetic energy from the axisymmetric vortex andpartly by the conversion of potential energy asso-ciated with convection (Wang, 2002a). Recently,Yang et al (2007) found that the asymmetries,such as those developed in our full physics ex-periment, play a negative role in tropical cycloneintensification and reduce the maximum potentialintensity. The results from this study seem tosupport their conclusions. As we can see fromFigs. 6 and 10, as the storm develops a strongPV annulus (Fig. 12) when it reaches its first peakintensity in maximum surface wind after 75 hof simulation, the wavenumber-one asymmetryamplifies considerably between 78 h and 84 h,reaching a maximum pressure perturbation of

Fig. 11. Time evolution in the amplitudes ofwavenumber-one, wavenumber-two, wavenum-ber-three, and the residual asymmetric pressure(hPa) in the full physics experiment

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 109

Fig. 12. The radial PV profiles of the azimuth-al mean storm at the initial and the given timesof simulation in the full physics experiment

Fig. 13. Radius-time Hovm€ooller diagram for the total (a), wavenumber-one (b), wavenumber-two (c), wavenumber-three (d),wavenumber-four, (e) and the residual asymmetric kinetic energy at 1 km height in the full physics experiment. Contourintervals are 2 m2 s�2 in (a) and 1 m2 s�2 in (b)–(f)

110 Y. Wang

about 10 hPa at 80 h of simulation. This largeamplification of the wavenumber-one asymmetryterminates the intensification and causes a tem-poral weakening of the storm (Fig. 6). Note thatthe asymmetric eddies play a critical role in mix-ing the PV between the eye and the eyewall, lead-ing to a less barotropically unstable PV structureof the azimuthal mean vortex (Fig. 12) andweaker asymmetries and thus a quasi–steady evo-lution of the storm (Fig. 6).

To examine the evolution of model asym-metries, we show in Figs. 13 and 14 the asym-metric kinetic energy at Z¼ 0.6 km and 12 km,respectively. The asymmetries are dominatedby low azimuthal wavenumber components andtrapped in the inner core region. This is consis-tent with the theoretical analysis of Smith andMontgomery (1995) and numerical results ofWang (2001; 2002a, b). They showed that thestrong shearing of the mean tangential wind of

the storm damps high wavenumber asymmetriesvery effectively, and leaves low wavenumber,quasi–balanced vortex Rossby waves in the innercore region. In both lower and upper troposphere,weak asymmetric eddies seem to be initiatedin the core region and then propagate radiallyoutward (Figs. 13a and 14a). Of most interest-ing is the periodic outward propagating signalsin eddy kinetic energy, in particular for bothwavenumber-one (Figs. 13a and 14a) and higherwavenumbers (Figs. 13f and 14f). Detailed ex-amination indicates that the wavenumber-oneasymmetries in the inner core could help initiateboth inner and outer spiral rainbands, which prop-agate radially outward and so do the wind anom-alies and thus the eddy kinetic energy associatedwith those spiral rainbands.

Although all the outward propagating epi-sodes look similar in the time-radial cross-sec-tion in Figs. 13 and 14, the first episode between

Fig. 14. The same as in Fig. 13, but at a height of 12 km. Contour intervals are 4 m2 s�2 in (a), 1 m2 s�2 in (b)–(e), and2 m2 s�2 in (f)

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 111

6 and 12 h is evidently quite different fromthe others. It occurs with convective outbursts(Fig. 10) and the associated convective down-drafts (not shown). The asymmetries are dom-inated by small-scale convective towers thatpropagate radially outward (Fig. 10). This is alsoreflected in the complicated wavenumber-one andwavenumber-two asymmetric structures during

this period shown in Figs. 8 and 9. It seems thatconvective outbursts are related to the convec-tive available potential energy (CAPE) whenthe azimuthal mean storm is barotropically sta-ble (Fig. 12). After 12–15 h of simulation, themean vortex develops an annular PV structure,which is barotropically unstable to the asym-metric perturbations. Such an annular PV struc-

Fig. 15. Total (top panels), wavenumber-one(middle panels), and wavenumber-two (bottompanels) asymmetries in pressure (left) andwinds (right) at 1 km height above the sea sur-face after 71 h of simulation in the full physicsexperiment. Contour interval is 1 hPa in pres-sure perturbations. The wind vector is scaled in10 m s�1

112 Y. Wang

ture seems to support the low azimuthal wave-number asymmetries in the inner core region(Figs. 8 and 9) where the radial PV gradientchanges its sign (Fig. 12). These low wavenum-ber asymmetries were identified as vortex Rossbywaves by Wang (2001; 2002a, b) and Chen andYau (2001). As discussed extensively in Wang(2001; 2002a), these waves are generally trapped

in the inner core region and their mass and windfields are quasi–balanced, namely low (high)pressure perturbation is collocated with cyclonicconvergent (anticyclonic divergent) flow. An ex-ample after 71-h simulation is shown in Fig. 15.These wave modes generally propagate cyclon-ically along the eyewall, as shown in Wang(2001; 2002a).

Fig. 16. Total (top panels), wavenumber-one(middle panels), and wavenumber-two (bottompanels) asymmetries in pressure (left) andwinds (right) at 1 km height above the sea sur-face averaged over the 120 h of simulation inthe full physics experiment. Contour interval is0.2 hPa in pressure perturbations. The windvector is scaled in 2 m s�1

A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4 113

A detailed examination indicates that the am-plitudes of these physical wave modes are slightlymodified by the computational asymmetries whilethey propagate along the eyewall. Since the nu-merical asymmetries have their preferred spatialstructure and are quasi–stationary as discussed forthe dry experiment in Sect. 3, we can easily iso-late these computational modes from the long-term (0–120 h) mean of the asymmetries in thefull physics experiment. Figure 16 shows such aresult, from which we can see that the quasi–sta-tionary component of the asymmetries in the fullphysics experiment is dominated by wavenumber-two component, which has a similar structureto that shown in Fig. 1 for the dry experiment.Note that although these computational stationarymodes are much stronger than those in the dryexperiment due to the much stronger storm inthe full physics experiment, they are still smallcompared to the corresponding non-stationaryasymmetric physical modes (e.g., Fig. 15). There-fore, the asymmetries of numerical origin mayplay a secondary role in affecting the storm struc-ture and intensity in the simulation. However,interpretation of asymmetries, especially the wave-number two asymmetries, in the inner core regionof a simulated tropical cyclone by numerical mod-els should be with caution.

In addition, as indicated in the introduction,some earlier three-dimensional numerical model-ing studies attributed the development of asym-metries to the inertial instability in the outflowlayer (Anthes et al, 1971; Anthes, 1972; Kuriharaand Tuleya, 1974). This was shown to happenafter the model tropical cyclone intensified strongenough to develop a strong anticyclonic outflowjet, creating regions with negative absolute vor-ticity (or negative PV) in the outflow layer. Sincethis type of negative PV was found to occur sev-eral hundred kilometers from the storm center,the inertial instability thus could not explain thedevelopment of asymmetries in the inner coreregion in our high-resolution model. In particu-lar, the asymmetries have their origin in the innercore region and then propagate radially outwardinstead of inward and they develop in the earlymodel integration when the outflow jet does notdevelop (Figs. 13 and 14). Even at later stages,we have not found any coherent negative PV

developed in the outflow layer. Therefore, thereis no any indication of inertial instability to play

a role in the initiation and development of asym-metries in our simulated storm.

5. Conclusions

A fully compressible, nonhydrostatic tropicalcyclone model – TCM4 has been developed tostudy the structure and intensity changes in trop-ical cyclones due to either internal dynamics orexternal forcing or both. TCM4 is an extensionof the previously developed hydrostatic modelTCM3 with the replacement of the hydrostaticdynamical core by a fully compressible, nonhy-drostatic dynamical core. TCM4 shares the state-of-the-art model physics, the two-way interactivemultiple nesting, and automatic mesh movementwith its hydrostatic counterpart TCM3. An ef-ficient forward-in-time, explicit time splittingscheme is developed for model integration withthe fifth-order upwind scheme which takes intoaccount the effect of spatial variation of the ad-vective flow for horizontal advection (Wang,1996). A major feature of TCM4 is its capabilityof simulating the inner-core structure and theassociated intensity change of a tropical cycloneat nearly cloud resolving resolution. The use ofmultiple nesting and automatic mesh movementin TCM4 allows us to use adequate medium sizesfor meshes with fine resolutions so that we cansave computer time. TCM4 also is a useful toolto study the interaction between a tropical cy-clone and its large-scale environmental flow.

In this study, we have documented TCM4 andused it to investigate how the asymmetric struc-ture develops in an initially axisymmetric tropicalcyclone without any explicit asymmetric forcing.The model was thus configured on an f-plane andinitialized with an axisymmetric cyclonic vortexin a quiescent environment with a constant seasurface temperature of 29 �C. Two experimentswere conducted: one with no model physics,named the dry experiment; and the other with fullmodel physics, named the full physics experi-ment. The results show that in the dry experiment,the initial development of asymmetries resultsfrom the numerical finite-differencing schemeon a regular square grid system and time-splittingerror associated with the horizontal advection.The asymmetries are dominated by azimuthalwavenumber-two and wavenumber-four struc-tures. Because the initial cyclonic vortex has a

114 Y. Wang

monotonic radial distribution of PV and thus isdynamically stable, the computational asymmet-ries in the dry experiment could not grow furtherafter their initiation and remain very weak andstationary, thus having little effect on the overallevolution of the dry vortex in the experiment.

In the moist experiment, the initial developmentof asymmetries is quite similar to that in the dryexperiment before the development of signifi-cant precipitation. However, once precipitation oc-curred with convective outbursts, the asymmetriesare dominated by small-scale convective cells=towers near the original radius of maximum wind.The rapid convective outbursts generate stronggravity waves that propagate radially outward.The convective outbursts also produce strongdowndrafts, lowering the equivalent potential tem-perature in the subcloud layer and leading to atemporal weakening of convection in the innercore. Although negative to the preexisting con-vection, the low equivalent potential temperaturepromotes strong surface entropy flux at the oceansurface due to the increased air-sea entropy defi-cits, providing a preconditioning for deep convec-tion in the inner core region and the subsequentrapid intensification of the storm. Convectiveheating associated with the inner core convectionproduces an off-center local PV maximum of theazimuthal mean vortex at a radius of about 20 kmin the mid-lower troposphere after about 12–15 hof simulation. The reverse of radial PV gradientacross this PV annulus satisfies the necessary con-dition for barotropic instability, resulting in arapid development of large amplitude asymmet-ries with low azimuthal wavenumbers.

The low azimuthal wavenumber asymmetriesin the inner core region are dominated by vortexRossby waves, as found in previous studies (Wang,2001, 2002a, b; Chen and Yau, 2001). It is alsoshown that these physical modes are slightlymodified by the quasi–stationary computationalasymmetries. Although the amplitude of the com-putational modes is relatively small compared tothe physical modes, interpretation of the asym-metries in the inner core region, especially thewavenumber-two component, at any individualtime, needs to be with caution. Further we foundno indication of contribution by inertial instabil-ity in the outflow layer to the development ofasymmetries in the inner core region of our simu-lated storm.

Acknowledgments

This study has been supported by NSF grant ATM-0427128and ONR grant 000-14-94-1-0493. Additional support hasbeen provided by the JAMSTEC through its sponsorshipof the International Pacific Research Center (IPRC) inthe School of Ocean and Earth Science and Technology(SOEST) at the University of Hawaii. This is the SOESTpublication 7063 and IPRC publication 433.

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Author’s address: Yuqing Wang, SOEST=IPRC, Univer-sity of Hawaii at Manoa, POST Bldg. Room 409G, 1680East-West Road, Honolulu, HI 96822, USA (E-mail: yuqing@hawaii.edu)

116 Y. Wang: A multiply nested, movable mesh, fully compressible, nonhydrostatic tropical cyclone model – TCM4