Potential Vorticity Diagnosis of a Simulated Hurricane ...dalin/wang-zhang-pvdiag1-formulation-s03.pdf1JULY 2003 WANG AND ZHANG 1593 q 2003 American Meteorological Society Potential

1 JULY 2003 1593W A N G A N D Z H A N G

q 2003 American Meteorological Society

Potential Vorticity Diagnosis of a Simulated Hurricane. Part I: Formulation andQuasi-Balanced Flow

XINGBAO WANG AND DA-LIN ZHANG

Department of Meteorology, University of Maryland at College Park, College Park, Maryland

(Manuscript received 20 August 2002, in final form 27 January 2003)

ABSTRACT

Because of the lack of three-dimensional (3D) high-resolution data and the existence of highly nonellipticflows, few studies have been conducted to investigate the inner-core quasi-balanced characteristics of hurricanes.In this study, a potential vorticity (PV) inversion system is developed, which includes the nonconservativeprocesses of friction, diabatic heating, and water loading. It requires hurricane flows to be statically and inertiallystable but allows for the presence of small negative PV. To facilitate the PV inversion with the nonlinear balance(NLB) equation, hurricane flows are decomposed into an axisymmetric, gradient-balanced reference state andasymmetric perturbations. Meanwhile, the nonellipticity of the NLB equation is circumvented by multiplying asmall parameter « and combining it with the PV equation, which effectively reduces the influence of anticyclonicvorticity. A quasi-balanced v equation in pseudoheight coordinates is derived, which includes the effects offriction and diabatic heating as well as differential vorticity advection and the Laplacians of thermal advectionby both nondivergent and divergent winds.

This quasi-balanced PV–v inversion system is tested with an explicit simulation of Hurricane Andrew (1992)with the finest grid size of 6 km. It is shown that (a) the PV–v inversion system could recover almost all typicalfeatures in a hurricane, and (b) a sizeable portion of the 3D hurricane flows are quasi-balanced, such as theintense rotational winds, organized eyewall updrafts and subsidence in the eye, cyclonic inflow in the boundarylayer, and upper-level anticyclonic outflow. It is found, however, that the boundary layer cyclonic inflow andupper-level anticyclonic outflow also contain significant unbalanced components. In particular, a low-level outflowjet near the top of the boundary layer is found to be highly unbalanced (and supergradient). These findings aresupported by both locally calculated momentum budgets and globally inverted winds. The results indicate thatthis PV inversion system could be utilized as a tool to separate the unbalanced from quasi-balanced flows forstudies of balanced dynamics and propagating inertial gravity waves in hurricane vortices.

1. Introduction

Nonlinear balanced (NLB) models have been widelyused in theoretical studies to help understand the fun-damental dynamics of tropical cyclones. The earliestwork could be traced back to Eliassen (1952) who de-veloped an axisymmetric nonlinear balance model toinvestigate how a hurricane vortex evolves under theinfluence of latent heat release and surface friction. Sub-sequently, various types of NLB models have been de-veloped to study the balanced characteristics of hurri-cane vortices (e.g., Sundqvist 1970; Challa and Pfeffer1980; Shapiro and Willoughby 1982). Balanced dynam-ics is of particular interest to many researchers becauseit enables one to identify and follow significant flowfeatures (in ‘‘slow manifold’’) in space and time.

Since the comprehensive review work of Hoskins et

Corresponding author address: Dr. Da-Lin Zhang, Department ofMeteorology, University of Maryland, College Park, MD 20742.E-mail: [email protected]

al. (1985), the potential vorticity (PV) concept has beenfrequently used for understanding the three-dimensional(3D) balanced dynamics of extratropical cyclones (e.g.,Reed et al. 1992; Huo et al. 1999a), tropical cyclones(e.g., Schubert and Alworth 1987), and convectivelygenerated midlevel mesovortices (Olsson and Cotton1997; Trier and Davis 2002). The PV concept is attrac-tive due to its conservative property in the absence ofdiabatic and frictional processes, and its invertibilityprinciple with which a complete 3D flow could be di-agnosed from a known PV distribution, given a balancedcondition and appropriate boundary conditions. How-ever, strong nonlinearity in the PV and NLB equations,when applied to mesoscale motion, makes it almost im-practical to perform the PV inversion. After applyingan ad hoc linearization, Davis and Emanuel (1991, here-after referred to as DE91; and later refined by Davis1992) were able to develop a PV inversion algorithmthat allows for decomposition of the 3D PV field in apiecewise manner for extratropical cyclones. This adhoc linearization appears to be necessary to absorb high-

1594 VOLUME 60J 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

order nonlinear terms and facilitate the convergence ofiterative calculations because their reference PV, deter-mined from its climatological (i.e., temporal mean) val-ue, and PV anomalies were at the same order of mag-nitude (e.g., see Zhang et al. 2002, their Fig. 6). Nev-ertheless, their PV inversion algorithm has successfullybeen used to diagnose the effects of condensational heat-ing (Davis and Weisman 1994; Davis et al. 1996), theinteraction of different PV anomalies (Huo et al. 1999a),the impact of removing upper-level perturbations on thesurface extratropical cyclogenesis (Huo et al. 1999b),the influence of upper- and low-level PV anomalies onthe movement (Wu and Emanuel 1995a,b), and the in-tensification of tropical cyclones (Molinari et al. 1998).

While DE91’s algorithm has been successfully usedby Wu and Emanuel (1995a,b) to diagnose hurricanemovement and the storm’s influence on its track, the PVanomalies associated with the hurricane vortex have tobe excluded in assessing the hurricane’s total advectiveflow due partly to the use of a climatological PV dis-tribution as a reference state and partly to the sensitivityof their inversion results to the choice of the hurricanecenter. Motivated by weak asymmetries in the hurricanecore, Shapiro (1996), and Shapiro and Franklin (1999)developed a PV inversion algorithm to study hurricanemotion by decomposing the horizontal winds into sym-metric (vortex) and asymmetric (environmental) com-ponents. Thus, the removal of hurricane vortices, asrequired by Wu and Emanuel (1995a,b), was avoided.

Based on the asymmetric balance (AB) theory of Sha-piro and Montgomery (1993), Moller and Jones (1998)developed a PV inversion algorithm to study the evo-lution of hurricane vortices in a primitive-equation mod-el, and later Moller and Shapiro (2002) used it to eval-uate balanced contributions to the intensification of Hur-ricane Opal (1995) in a Geophysical Fluid DynamicsLaboratory (GFDL) model forecast. The AB theory re-duces to Eliassen’s formulation for purely axisymmetricflow. An advantage of the AB theory is that it provideslittle restriction on the magnitude of divergence (Mont-gomery and Franklin 1998; Moller and Shapiro 2002),as compared to the small divergence assumed in theNLB models. However, the validity of the AB theorydepends on a nondimensional parameter, that is, ( /2Dl

Dt2)/( ) K 1, which is the ratio of the azimuthal tohjinertial frequencies in a symmetric vortex, where Dl/Dt 5 ]/]t 1 Vt/r)]/]l, Vt is the mean tangential wind,and (r, l) denotes the (radius, azimuth) axes of thecylindrical coordinates, and 5 ( f 1 ]Vt/]r 1 Vt/r)his the vertical component of mean absolute vorticity and

5 f 1 2Vt/r is the inertia parameter. A scale analysisjby Shapiro and Montgomery (1993) indicates that thevalidity of the AB theory requires the squared localRossby number

2 2 2n V /rt2R 5 K 1,nhj

when /Dt2 is evaluated by the advective contribution2Dl

n2 /r2, where n is the azimuthal wavenumber. This2V t

implies that the AB theory may not be very accuratefor high wavenumber asymmetries, even though theabove scaling may have somewhat overestimated themagnitude of /Dt2, according to Moller and Mont-2Dl

gomery (1999) and Moller and Shapiro (2002).Despite the above-mentioned advantage of the AB

theory, the NLB models or similar balance algorithmshave been widely used, due to their relative simplicities,to diagnose the balanced flows associated with meso-scale convective systems, including intense hurricanevortices (McWilliams 1985; Zhang et al. 2001). More-over, the divergent component of 3D flows may be ob-tained from the balanced winds through the quasi-bal-anced v equation (Davis et al. 1996; Olsson and Cotton1997). Nevertheless, all of the balance formulationsmentioned above set PV to be a positive threshold valueeverywhere in order to satisfy an ellipticity requirementfor finding a solution. In fact, PV is often not positivein the upper outflow layer of hurricanes where intenseanticyclonic vorticity is present. In addition, because ofthe lack of high-resolution, high quality winds and ther-modynamic data, most of the previous PV diagnosticstudies focused on the broad-scale balance character-istics of hurricane vortices, and few studies have beenperformed to investigate the quasi-balanced asymmetriccharacteristics in the inner core regions of a hurricane.

In this study, a PV inversion system is developed,following DE91, but their ad hoc linearization is aban-doned and the ellipticity requirement of the NLB equa-tion is circumvented. This PV inversion system will thenbe tested using a high-resolution (Dx 5 6 km) explicitsimulation of Hurricane Andrew (1992) with the fifth-generation Pennsylvania State University–NationalCenter for Atmospheric Research (PSU–NCAR) non-hydrostatic mesoscale model (i.e., MM5). Liu et al.(1997, 1999) have shown that the triply nested grid (54/1816 km) version of MM5 reproduces reasonably wellthe track, intensity, as well as the structures of the eye,eyewall, spiral rainbands, the radius of maximum wind(RMW), and other inner-core features as compared toavailable observations and the results of previous hur-ricane studies. Their simulation provides a complete dy-namically and thermodynamically consistent dataset forus to examine the feasibility and accuracy of this PVinversion system in diagnosing the quasi-balancedasymmetric characteristics in the storm’s inner-core re-gions.

The next section shows the derivation of the PV in-version system, including the PV, NLB, and v equationsin pseudoheight Z coordinates. The associated ellipticconditions will be discussed. Section 3 describes com-putational procedures of the inversion algorithm as ap-plied to a model-simulated hurricane vortex. Section 4presents the inverted (axisymmetric and asymmetric)results as verified against the model-simulated. Somebalanced and unbalanced characteristics of hurricane


FIG. 1. Radial distribution of the wavenumber-0 (solid), wavenum-ber-1 (dashed), wavenumber-2 (dotted), and wavenumber-3 (dot-dashed) components of tangential winds at an altitude of 1 km, thatare obtained by averaging 15 model outputs at 4-min intervals duringthe 1-h period from the 56–57-h simulation ending at 2100 UTC 23Aug 1992.

flows will be shown. A summary and concluding re-marks will be given in the final section.

2. Formulation

a. Potential vorticity inversion equations

Because of the importance of water loading in hy-drostatic balance in the eyewall (Zhang et al. 2000), itis appropriate to begin with a more general form ofErtel’s PV that takes into account of the moisture andprecipitation effects (see Schubert et al. 2001):

1Q 5 h · =u , (1)rr

where r is the total density consisting of the sum of thedensities of dry air, airborne moisture (vapor and cloudcondensate), and precipitation; h 5 2V 1 = 3 V isthe absolute vorticity vector; and ur is the virtual po-tential temperature with the water loading effect in-cluded (see appendix A for its definition). Schubert etal. (2001) showed that if a flow is balanced, there existsan invertibility principle with Eq. (1) that could yieldthe balanced mass and wind fields from the spatial dis-tribution of PV. In the present study, the NLB equationis used as our balance condition. To derive the NLBequation, the horizontal wind Vh is decomposed intonondivergent and irrotational components by defining astreamfunction C, and a velocity potential X; that is,

V 5 (u, y) 5 2=C 3 k 1 =X 5 V 1 V ,h C X (2)

where | VX | K | VC | , and C and X satisfy the followingrelations, respectively:

2 2¹ C 5 k · = 3 V 5 z, ¹ X 5 = · V 5 D, (3)h h

where z is the vertical component of relative vorticity,and D is the horizontal divergence. Given z and D, Cand X can be inverted from (3) with appropriate bound-ary conditions.

With the above definition, the NLB equation relatingthe horizontal wind to the mass field can be written as

]C ]C2¹ F 5 = · (F= C) 1 2J , 1 = · F , (4)h h h h r1 2]X ]Y

where the capital letters denote dimensional variables,=h is a 2D (X, Y) gradient operator, F is the geopotentialheight, F is the Coriolis parameter, J is a Jacobian op-erator, and Fr represents the frictional force includingthe effects of the numerical diffusion and the planetaryboundary layer (PBL). Equation (4) is generally validwhen the divergent component VX is much weaker thanthe rotational component VC. This relation could beapplied to a flow with large curvature, provided that theFroude number and aspect ratio are small (McWilliams1985). Note that for an axisymmetric vortex on an Fplane, Eq. (4) reduces to the gradient wind balance(GWB) in the absence of friction. Previous observa-tional and modeling budget studies showed that GWB

is a good approximation to the azimuthally averagedwinds above the PBL within a 10% error (Willoughby1990; Zhang et al. 2001).

Under the hydrostatic and nondivergent assumption,Eq. (1) can be rewritten as

2 2 21 u ] F ] C ] F0 2Q 5 (F 1 ¹ C) 2h 2[r(Z ) G ]Z ]X]Z ]X]Z

2 2] C ] F2 , (5)]]Y]Z ]Y]Z

where Z 5 [1 2 (P/P0) ](Cpu0/G) represents the ver-R/Cp

tical pseudoheight coordinate, G is dimensional gravi-tational acceleration, and r(Z) 5 r0(P/P0) is pseu-C /Cy p

dodensity (Hoskins and Bretherton 1972); see appendixA for a closed set of dynamical equations in Z coor-dinates used in this study. Given a PV distribution, Eqs.(4) and (5) are a closed set of equations in F and Cthat may be solved iteratively to find a solution. How-ever, several issues need to be considered before sucha solution can be obtained.

First, Eqs. (4) and (5) could be highly nonlinear, de-pending on their applications, so a unique solution maynot always be obtainable. Thus, it is desirable to line-arize them by decomposing the total hurricane flow intoa reference state and a perturbation component, as dem-onstrated by DE91 for extratropical cyclones. Prior tothe decomposition, one should note that the balancedsolution from Eqs. (4) and (5) requires the flow to beinertially and statically stable (Hoskins et al. 1985), butthe snapshot model fields do not always satisfy the sta-bility requirements. Therefore, to ensure the inertial andstatic stability of hurricane flow, we (a) set the verticalcomponent of absolute vorticity h to be slightly positive(i.e., h $ 0.1 f ); and (b) adjust the vertical ur structure


such that ]ur/]Z $ 0.01 N 2u0/G is always held, whereN 2 (51024 s21) is the buoyancy frequency, using thevariational method of Sasaki and McGinley (1981). Itis found that procedure (a) tends to be activated mostlyin the upper outflow layer, whereas procedure (b) takesplace mostly in the boundary layer and aloft in the eye.Of importance is that the present procedures do notrequire Q . 0 because of the possible existence ofnegative product of horizontal vorticity and ur gradient[see Eq. (5)]. However, after the above procedures, wedid find that negative PV becomes weaker in the upperoutflow layer and this tends to yield positive PV whenit is azimuthally averaged.

To facilitate the subsequent discussions, it would bemore convenient to rewrite Eqs. (4) and (5) in nondi-mensional form (see appendix B for more details); thatis,

]c ]c2¹ f 5 = · ( f = c) 1 2J , 1 = · f , (49)h h h h r1 2]x ]y

2 2 2 2 2] f ] c ] f ] c ] f2q 5 ( f 1 ¹ c) 2 1 ,h 2 1 2[ ]]z ]x]z ]x]z ]y]z ]y]z

(59)

where the lower cases denote nondimensionalized var-iables, unless mentioned otherwise hereafter. Note thateach term in the above equations has one-to-one cor-respondence to that in Eqs. (4) and (5).

Because of the dominant axisymmetric nature of hur-ricanes, we decompose the hurricane flow fields intoaxisymmetric and its deviation (asymmetric) parts. Indoing so, we first define the reference geopotentialheight as its azimuthal average centered at the min-fimum surface pressure. The nondimensionalized refer-ence variables and are then obtained from the GWBc qrelation and Eq. (59), respectively. An advantage of thisprocedure is that we do not need to iteratively solveEqs. (49) and (59) for and , given . [We have triedc f qto calculate first from the model output and thencsolved Eq. (49) as Poisson equation for . But the re-fsulting leads to negative static stability near the topfof the PBL where the tangential wind is peaked.] Allthe nondimentionalized perturbation variables are thenobtained by

c9 5 c 2 c , f9 5 f 2 f, and

q9 5 q 2 q. (6)

Figure 1 shows the radial distribution of azimuth-ally averaged tangential winds Vt (i.e., wavenumber0) and their perturbation amplitudes in wavenum-V 9tbers 1–3 at an altitude of 1 km where the tangentialwinds are nearly maximized. The tangential wave-numbers 1–3 winds are obtained after performing az-imuthal Fourier decomposition in cylindrical coor-dinates. Obviously, the perturbation wind componentsare about one order of magnitude smaller than the

azimuthal mean in the eyewall and outer regions (i.e.,c 9 K , and f9 K ), unlike in the previous extra-c ftropical cyclogensis studies in which the referenceand perturbation states were at the same order of mag-nitude when temporal means were used. Moreover,the higher the wavenumber, the smaller is the mag-nitude of perturbation winds. These results are similarto those shown in Shapiro and Montgomery (1993),and they also conform to the observational findingsof Marks et al. (1992), Franklin et al. (1993), andReasor et al. (2000) that the asymmetric componentsare less than 20% of the symmetric component. Thus,decomposing the total hurricane flow into the refer-ence and perturbation components is consistent withthe basic assumption involved in linearizing the non-linear equations of (49) and (59), as given below.

Substituting (6) into Eqs. (49) and (59) gives

]c92 2¹ f9 5 f ¹ c9 1 bh h ]y

2 2 2 2 2 2] c ] c9 ] c ] c9 ] c9 ] c1 2 2 2 1

2 2 2 21 2]x ]y ]x]y ]x]y ]x ]y

22 2 2] c9 ] c9 ] c9 ] f ] f yx1 2 2 1 1 , (7)2 2 1 2[ ]]x ]y ]x]y ]x ]y

and2 2 2 2] f9 ] f ] c ] f9

2 2q9 5 ( f 1 ¹ c ) 1 ¹ c9 2h h2 2]z ]z ]x]z ]x]z2 2 2 2 2 2] c9 ] f ] c ] f9 ] c9 ] f

2 2 2]x]z ]x]z ]y]z ]y]z ]y]z ]y]z

2 2 2 2 2] f9 ] c9 ] f9 ] c9 ] f921 ¹ c9 2 2 , (8)h 2]z ]x]z ]x]z ]y]z ]y]z

where b 5 ] f /]y. Although there are still a few non-linear terms involving P (c9, f9) and P (c9, c9) onthe right-hand side (rhs) of the above equations, theyare one order of magnitude smaller than the linear terms,based on the results given in Fig. 1. Thus, the nonlin-earity of Eqs. (7) and (8) is much weaker than that usedto study extratropical cyclones. Furthermore, it can bereadily shown that all these nonlinear terms will be ab-sorbed if we follow DE91 by inserting c* 5 1 0.5c9cand f* 5 1 0.5f9 into Eqs. (7) and (8).f

Second, if we solve (7) for c9 with f9 obtained from(8), Eq. (7) is an elliptic equation of c9 such that itsassociated elliptic condition must be met in order to finda solution (Arnason 1958). In the case of a stationarycircular vortex, such a condition, given in cylindricalcoordinates, can be written as

V ]Vt tf 1 2 f 1 2 . 0. (9)1 21 2R ]R

The above criterion restricts the magnitude of anticy-clonic shear vorticity ]Vt/]R. Our calculations with the


simulated Andrew indicate that a sizeable area outsidethe RMW above the PBL is nonelliptic due to the pres-ence of strong anticyclonic shear [i.e., ( f 1 2(]Vt/]R))

, 0]. To circumvent the ellipticity requirement, we mul-tiply Eq. (7) by a parameter «, varying between 0 and1, and then add it into Eq. (8) to obtain

2 2 2 2 2 2 2 2 2 2 2] f ] f9 ] c ] f9 ] c9 ] f ] c ] f9 ] c9 ] f ] f92 2 2« f 1 ¹ c9 5 q9 2 ( f 1 ¹ c ) 1 1 1 1 2 ¹ c9h h h2 2 21 2]z ]z ]x]z ]x]z ]x]z ]x]z ]y]z ]y]z ]y]z ]y]z ]z

2 2 2 2 2 2 2 2 2 2] c9 ] f9 ] c9 ] f9 ]c9 ] c ] c9 ] c ] c9 ] c9 ] c21 1 1 « ¹ f9 2 b 2 2 2 2 1h 2 2 2 21 2[]x]z ]x]z ]y]z ]y]z ]y ]x ]y ]x]y ]x]y ]x ]y

2 2 2 2] c9 ] c9 ] c9 ] c ] f ] f yx2 2 2 2 2 . (10)2 21 2 ]]x ]y ]x]y ]x]y ]x ]y

Since the nonlinear terms P (c9, f9) and P (c9, c9)in the above equation are much smaller than the linearterms, we may assume the following elliptic conditionfor (10) approximately,

2 2 2 2] c ] f ] c ] f« f 1 2« 1 « f 1 2« 1

2 2 2 21 21 2]x ]z ]y ]z

22] c2 4« . 0. (99)1 2]x]y

where ]2 /]z2(5 2) is equivalent to the mean staticf Nstability. It is evident from (99) that the value of « tobe used should be as small as possible, unless the at-mosphere is very stable (e.g., in winter storms). How-ever, « cannot be too small due to its presence in thecoefficient (i.e., « f ) on the lhs of (10) in cases of weakstatic stability. In fact, we found that « 5 0 often leadsto the divergence of iterative calculations, when solvingfor c9 from (10), at points where the static stability isrelatively weak or q9 is large. On the other hand, whenthe vertical adjustment criterion of ]ur/]Z $ 0.2 N 2u0/G is used, any value of « between 0 and 1 could resultin a convergent solution. For the criterion chosen herein(i.e., ]ur/]Z $ 0.01 N 2u0/G), the range of « 5 0.1 ;0.8 would yield a solution without encountering anynonellipticity problem. Thus, in this study a compromisevalue of « 5 0.5 is adopted to reduce the influence ofanticyclonic shear vorticity on the elliptic requirementwhile enhancing the effects of static stability on theinduced circulations associated with a given q9 distri-bution.

Subtracting (8) from (7) leads to2] f9

2 2¹ f9 1 ( f 1 ¹ c )h h 2]z2] f ]c9 ] f ] f yx2 25 q9 2 ¹ c9 1 f ¹ c9 1 b 1 1h h2]z ]y ]x ]y

2 2 2 2 2 2 2] f9 ] c ] f9 ] c9 ] f ] c ] f922 ¹ c9 1 1 1h 2]z ]x]z ]x]z ]x]z ]x]z ]y]z ]x]z

2 2 2 2 2 2] c9 ] f ] c9 ] f9 ] c9 ] f91 1 1

]y]z ]x]z ]x]z ]x]z ]y]z ]y]z

2 2 2 2 2 2] c ] c9 ] c ] c9 ] c9 ] c1 2 2 2 1

2 2 2 21 2]x ]y ]x]y ]x]y ]x ]y

22 2 2] c9 ] c9 ] c91 2 2 . (11)

2 2 1 2[ ]]x ]y ]x]y

Since perturbation variables are much smaller than theirreference values, it is important to realize that the so-lution obtained from (10) and (11) is superposable tothe reference state—an issue raised by Bishop andThorpe (1994) and Thorpe and Bishop (1995).

Third, solving Eqs. (10) and (11) requires appropriatelateral, top and bottom boundary conditions. From thehydrostatic equation

]f g5 u , (12)r]z u0

Neumann boundary conditions for f9 can be specifiedat the bottom (z 5 zB) and top (z 5 zT) boundaries,respectively, as follows:

]f9 g ]f9 g5 u9 ; 5 u9 . (13)rB rT) )]z u ]z u0 0z5z z5zB T

Dirichlet boundary conditions for c9 and f9 are spec-ified along the lateral boundaries, where f9 matches thesimulated or observed, and the tangential c9 gradientmatches the normal component of horizontal winds sim-ilar to that used by DE91. That is,

V · n dlhR]c9

5 2V · n 1 , (14)h]sdlR

where l is a path along the lateral boundaries, s is avector tangent to that path, and n a vector normal to


that path. Note that the domain-averaged net divergence,for example, resulting from the low-level inflow andupper-level outflow of hurricanes, must be deducted inorder to satisfy the basic assumptions in deriving Eq.(7); so the second term is included on the rhs of Eq.(14). Of course, this term can be excluded when a largedomain is used, for example, by Wu and Emanuel(1995a,b) and Shapiro (1996).

b. An v equation for a quasi-balanced model

The PV inversion discussed above will yield 3D (non-divergent) balanced mass and horizontal wind fields thatcontain little vertical motion or secondary circulationsof hurricanes. However, the secondary circulations, con-sisting of convergent inflow in the PBL, slantwise ver-tical motion in the eyewall, and divergent outflow inthe upper troposphere, play an important role in deter-mining the intensity of hurricane vortices and the de-velopment of clouds and precipitation in the eyewall.Clearly, any balanced theory of hurricanes would beincomplete without including the divergent wind com-ponent. However, including the divergent component toa balanced flow would differ from its conventional bal-ance definition. In this regard, it would be more appro-priate to consider the resulting total 3D flow as beingquasi-balanced, because this divergent component as-

sociated with the balanced flow is presumably in ‘‘slowmanifold,’’ unlike the divergent flow associated withgravity waves that tends to radiate away from energysources.

Secondary circulations in hurricanes may result fromthe balanced flow (i.e., c and f) and any dynamical andphysical processes (e.g., friction fr or diabatic heatingqr), which can be determined by an v equation andmass continuity equation, in a manner similar to theirquasigeostrophic analog. To show this, we start fromthe vertical vorticity z equation

]z ]h5 2(V 1 V ) · = h 2 v 2 h= · Vc x h h x]t ]z

]v ]u ]v ]y ] f ] fy x1 2 1 2 , (15)]y ]z ]x ]z ]x ]y

and the thermodynamic equation

]u ]u ]u ]ur r r r1 u 1 y 1 v 5 q , (16)r]t ]x ]y ]Z

where v is vertical velocity (positive implies upwardmotion) in pseudoheight z coordinates (see appendix A).Performing (g/u0) [Eq. (16)] 2 ]2 [Eq. (49)]/]t]z 22¹h

f] [Eq. (15)]/]z, and after some manipulations, we ob-tain the following quasi-balanced v equation

2 2 2 2 2] f ] ] ] ]v ] c ]v ] c ] ]v ] x ]v ] x2 2m m¹ v 1 f h (z 2 z) [(z 2 z) v] 2 f 1 2 f 2h a a21 2 5 6 1 2 1 2]z ]z ]z ]z ]x ]x]z ]y ]y]z ]z ]x ]y]z ]y ]x]z

2 2 2 2 2 2 3]h m ] h ] ]f ] ] c ] c ] c ] c ] c22 f 1 f v 5 f [V · =h] 2 ¹ V · = 2 2 2 2 bh h h h2 2 21 2 1 2[ ]]z z 2 z ]z ]z ]z ]t]z ]x ]y ]x]y ]x]y ]t]y]za

2g ] ] f ] f ] ] f ] fy yx x21 ¹ q 2 f 2 2 1 , (17)h r 1 2 1 2u ]z ]x ]y ]t]z ]x ]y0

where m 5 Cy /Rd, c, and f are the total streamfunctionand geopotential height. The dimensional form of Eq.(17) is similar to that derived by Krishnamurti (1968)and DE91 except for the use of z coordinates and in-clusions of the velocity potential x, and the frictionaland water loading effects. Apparently, the vertical mo-tion of the quasi-balanced model can be determined bythe rhs forcing terms of Eq. (17), which, from the leftto right, are the differential vorticity advection and theLaplacians of thermal advection both by nondivergentand divergent winds, the differential deformation or Ja-cobian term, the b effect, latent heating, and the effectsof friction, respectively. The first four forcing terms aremore dynamically related and diagnosed from the NLB–PV equation model; in these four terms, divergent windsalso play an important role in advecting various flow

properties (Raymond 1992). The last three terms aremore physically related and the major sources in de-termining the magnitude of divergent winds and sec-ondary circulations in hurricanes, which will, in turn,contribute to the intensities of the first four dynamicforcing terms.

Apparently, Eq. (17) is not a fully diagnostic equationbecause of the existence of two local tendency terms;that is, we must know ]c/]t, ] f x/]t, and ] f y/]t in orderto find v. The term ]c/]t could be obtained by eithersolving Eq. (15), like in Krishnamurti (1968), or takinga time derivative of Eqs. (4) and (5), like in DE91; theformer approach is adopted here. The frictional termsof ] f x/]t and ] f y/]t are obtained directly from the modeloutput.

Because of an additional variable x included in Eq.


(17), which will give the divergent, radial componentof hurricane vortices, we need to introduce the conti-nuity equation

]2 2m m¹ x 5 2(z 2 z) [(z 2 z) v]. (18)a a]z

Thus, if the diabatic and frictional contributions areknown (e.g., from a model simulation), Eqs. (15), (17),and (18) form a closed set of equations in v, x, and ]c/]t.

In summary, the two sets of equations [i.e., Eqs. (10),(11) and (15), (17), (18)] plus the reference state definecompletely a 3D quasi-balanced flow field (both non-divergent and divergent) through c, x, and v, and aquasi-balanced mass field through f. From these quan-tities, all the other relevant balanced properties of thefluid system (e.g., temperature) can be deduced. Sincethe two sets of equations are in forcing and responsemode, the piecewise partitioning and inversion could bein principle conducted to gain insight into the impactof various isolated PV perturbations or different phys-ical processes on hurricane flows.

3. Computational procedures

To demonstrate how well the inner-core quasi-bal-anced mass and winds of a hurricane could be invertedfrom a given PV distribution using the algorithm de-veloped in the preceding section, the model-simulatedHurricane Andrew (1992) data are output over the twofine-mesh (18 and 6 km) domains at 4-min intervalsfrom the 56–57-h integration, valid at 2000–2100 UTC,23 August 1992. During this period, the storm has en-tered its mature stage with a (near steady) maximumsurface wind of 68 m s21, although its central pressureis still decreasing at a rate of 1 hPa h21 (see Liu et al.1997, their Fig. 2). All the snapshot mass and windfields, as well as the forcing (diabatic and friction) andlocal tendency terms, are then decoupled from theirMM5 flux forms.

To reduce the influence of lateral boundaries on thefinal solution [e.g., see Eq. (14)], the finest (6 km) res-olution domain is extended in the y direction by inter-polating the intermediate (18-km resolution) mesh data,leading to the (x, y) dimensions of 124 3 124 pointscovering a horizontal area of 738 km 3 738 km. Themodel data are then interpolated in the vertical fromMM5’s 23 uneven s layers to 34 even z coordinateslayers to facilitate the PV inversion using the multigridsolver MUDPACK of Adams (1993). Note that the 34(even) layers could not retain the original high verticalresolution model data in the PBL. As will be seen inthe next section, the reduced vertical resolution tendsto misrepresent somewhat the boundary layer frictionaleffects; see Liu et al. (1997, 1999) for a detailed de-scription of the model configurations.

To decompose the simulated hurricane flow into areference (axisymmetric) state and a perturbation state,

all the 3D data are transformed from the (x, y, z) to thecylindrical (r, l, z) coordinates with its origin definedat the minimum sea level pressure of the storm, follow-ing the procedures described in Liu et al. (1999), andthen averaged azimuthally to yield the GWB referencestate. All the reference and perturbation variables arenondimensionalized. With 3D perturbation PV (q9) fromthe model output, Eqs. (10) and (11) are iterativelysolved, subject to the boundary conditions (13) and (14),for the perturbation mass and winds; the results are thenadded to their reference state. After obtaining the bal-anced total mass and winds, Eqs. (15), (17), and (18)are iteratively solved with the diabatic and frictionalforcing terms from the model output. In doing so, wedefine P 5 980 hPa as the bottom boundary in thepresent z coordinates, which puts the eye region (withthe minimum central pressure of less than 925 hPa)essentially above the sea surface. Clearly, in this situ-ation, it is not appropriate to specify v 5 0 as the bottomboundary condition. Thus, the mass continuity equation(18) is integrated from the model top downward, fol-lowing Olsson and Cotton (1997), assuming v | 5z5zr

0. Finally, all the inverted fields are averaged temporallyover a 1-h period, and then compared to the hourlyaveraged model outputs that are similar in magnitudeand structure to those shown in Liu et al. (1999) andZhang et al. (2001).

4. Quasi-balanced flow

In this section, the quasi-balanced hurricane mass andwind fields are inverted with the PV inversion systemdeveloped in section 2, and then verified against themodel simulated in both the west–east and the azi-muthally averaged radius–height cross sections, in orderto examine its capability in representing the quasi-bal-anced, asymmetric and axisymmetric structures of inner-core flows. Figure 2 compares the simulated PV and in-plane flow vectors to the inverted in west–east verticalcross sections. Pronounced asymmetries of the eyewalland its associated flow features are apparent (Fig. 2a).For example, the western eyewall is characterized bymore intense upward motion and upper-level outflowwith a deeper secondary circulation than those in theeast. This is consistent with the development of moreintense deep convection and precipitation in the westsemicircle (see Liu et al. 1997, their Fig. 5b). Of interestis the concentration of high PV (over 40 PVU; 1 PVU5 1026 m2 K kg21 s21) in the eye where little latentheating occurs; it coincides with the location of an in-tense thermal inversion layer near z 5 2.5 km. Thisfeature will have to be studied through PV budget cal-culations and it will be reported in a future article. Offurther interest is the development of a U-shaped PVstructure above the warm core (i.e., near z 5 6 km) thatappears to result from the cyclonic slantwise upwardadvection of the high PV associated with latent heatrelease in the eyewall and the slow cyclonic downward


FIG. 2. West–east vertical cross sections of PV (contoured), su-perposed with storm-relative in-plane flow vectors, from (a) the modeloutput (every 5 PVU), (b) the inverted or recalculated (every 5 PVU),and (c) the differences (every 0.5 PVU) between (a) and (b) [i.e., (a)minus (b)]. They are obtained by averaging 15 datasets at 4-minintervals during the 1-h period ending at 2100 UTC 23 Aug 1992.Shadings denote the simulated radar reflectivity greater than 15 and35 dBZ, which represents roughly the distribution of precipitationwith two different intensities. Solid (dashed) lines are for positive(negative) values. Note that vertical velocity vectors have been am-plified by a factor of 5.

advection of low PV in the eye. Negative PV is visiblein the upper-level outflow where anticyclonic vorticityis large.

It is encouraging from Figs. 2b and 2c that the re-calculated PV from the inverted mass and wind fieldsand the inverted in-plane flows compare favorably tothe simulated, including the magnitude and distributionof PV, and the storm’s secondary circulations. In par-ticular, the peak PV in the eye’s inversion layer (andanother PV center below), which appears to dominatethe hurricane vortex, is well recovered, indicating thegreat capability of this inversion technique in recoveringthe hurricane flows. However, some differences in thePBL and the upper outflow layer are notable. They couldbe attributed partly to the use of reduced vertical res-olution (i.e., in representing the frictional effects) andof the required nonnegative absolute vorticity and largerstatic stability, respectively. Nevertheless, the upper-lev-el anticyclonic outflow might be in a transition stage incase symmetric instability (i.e., PV , 0) is operative.

Figure 3 compares the inverted primary circulationof the hurricane to the simulated. Evidently, the PVinversion recovers the basic structures and magnitudesof the hurricane’s tangential flows, such as weak flowsin the eye, intense asymmetric wind maximums (over65 m s21) near the top of the PBL, decreasing windsin the PBL, and the slantwise sloping RMW with largenegative vertical shears in the eyewall (cf. Figs. 3a,b).As simulated, the inverted RMW axis is also situatedoutside of the updraft core. However, the globally in-verted fields are generally smoother than the simulated(e.g., wavy isotachs and locally reversed vertical shearsnear z 5 2 km and R 5 40 km), as expected. Again,more notable differences appear in the PBL where thedynamic unbalance is more pronounced due partly tothe surface friction and partly to the previously men-tioned inaccurate representation of the frictional effects,and in the upper levels where the inverted flows aremore constrained by rotational dynamics due to the re-quirement of nonnegative absolute vorticity (Fig. 3c).In addition, slightly stronger tangential flows (65 ms21) are inverted in the vicinity of the eye–eyewall in-terface where significant downdrafts appear. Despite thestated limitations and some possible inversion and in-terpolation errors, most of these differences could beconsidered as the unbalanced component of the hurri-cane flows, especially for the layers above the PBLwhere the differenced field is relatively small.

Of significance is that the unbalanced tangentialwinds become much weaker after the azimuthal average(see Fig. 4c), suggesting that most of them may be as-sociated with inertial gravity waves as those shown byLiu et al. (1999) since the magnitudes of other possiblewaves (e.g., acoustic) are small. This remains a subjectfor future study. In particular, the inverted radius–heightdistribution of tangential flows compares extremely wellwith the simulated (cf. Figs. 4a,b). The globally invertedunbalanced flows (i.e., 62–4 m s21), though differing


FIG. 3. As in Fig. 2, but for tangential winds contoured at intervalsof (a), (b) 5 m s21; and (c) contoured at 0, 62.5, 65, 67.5, and 620m s21. Solid (dashed) lines denote tangential flows [in (a) and (b)]or their differenced flows [in (c)] into (out of ) the page.

FIG. 4. As in Fig. 3, but for height–radius cross sections of tan-gential winds within the radius of 150 km. Note that (c) isoplethsare given at 0, 62.5, 65.0, 610, and 615 m s21.

in structure, are also similar to the locally derived agra-dient winds above the PBL (cf. Fig. 7 in Zhang et al.2001 and Fig. 4c herein). These results all indicate thatthe azimuthally averaged tangential flows above thePBL are approximately in GWB, conforming to the ob-servational analysis of Willoughby (1990) and the mo-mentum budget study of Zhang et al. (2001). The in-verted large positive (negative) unbalanced tangentialwinds (about 12 m s21) associated with a low-level ra-

dial outflow jet are also consistent with those (about 16m s21) shown in Zhang et al. (2001, cf. their Fig. 7b,and Fig. 4c herein). We speculate that this unbalancedcomponent would be much weaker if the mass field isforced toward the wind field. That is, the reference state

is calculated first and then Eq. (49) is solved as Poissoncequation for . As mentioned earlier, this procedureftends to generate statically unstable lapse rates near thetop of the PBL.


FIG. 5. As in Fig. 2, but for vertical motion at intervals of 0.4 ms21. Dark shadings denote latent heating rates greater than 10 and 30K h21, whereas light shadings denote latent cooling rates less than20.5 and 23 K h21.

The inverted secondary circulations from the quasi-balanced v equation (17) are compared to the simulatedin Fig. 5, which shows well the development of deep,boundary-layer-rooted and shallow, ‘‘suspended’’ slant-wise updrafts in the western and eastern eyewall, re-spectively. Of significance is that both the inverted andsimulated vertical motion bands are almost in phase withlatent heat release, especially in the eyewall and majorrainbands where more intense latent heating occurs. The

inverted maximum updrafts are about 2.0 and 1.0 m s21

in the respective western and eastern eyewall, as com-pared to the simulated 3.5 and 1.6 m s21 (cf. Figs. 5a,b),indicating that about 60%–70% of peak updrafts in theeyewall are quasi-balanced. The updraft and downdraftbands in the outer regions are also reasonably recovered,albeit with different intensities from the simulated.Thus, we may state that the quasi-balanced v equationcan recover reasonably the vertical motion field of ahurricane for either diagnostic analyses or model ini-tialization, given 3D latent heating and friction alongwith its vortex flow. After all, vertical motion in hur-ricanes represents the response of a primitive equationor balanced model to diabatic heating and the PBL ef-fects.

Of importance is that the Laplacians of latent heatingin the eyewall and other dynamical processes, as shownin Eq. (17), yield general (balanced) subsidence in theeye where little physics forcing occurs. However, theupper-level subsidence appears to be overestimated, ascompared to the simulated; this is consistent with thereduced anticylonic outflow and the increased returnflow into the eye in the inverted fields (cf. Figs. 5b and3c). In contrast, the pronounced downdrafts, originatingfrom the upper-level return flow and coinciding with theevaporative cooling at the eye–eyewall interface, do notappear in the inverted field (cf. Figs. 5a,b). This mightbe a result of the horizontal resolution used (i.e., Dx 56 km) that is too coarse for the model integration andPV inversion to resolve the interface downdrafts radi-ally. On the other hand, these narrow downdrafts arefound to be driven mostly by cold advection in radialinflow (see Zhang et al. 2002), so they are not deemedto be possibly inverted from the given weak coolingrates. Of further importance is that propagating gravitywaves shown in Liu et al. (1999, see their Fig. 8) arecompletely absent in Fig. 5b or at any other time (notshown). Instead, these waves appear in the differencedv field, in both the inner core and outer regions (Fig.5c); some v bands are 908 out of phase with the latentheating. These bands are unbalanced by definition andmust be consistent with their pertinent unbalanced windand mass fields.

Figure 6 compares the height–radius cross sectionsof azimuthally averaged, inverted vertical motion to thesimulated. Again, the v equation reproduces a robustupdraft in the eyewall and general subsidence in the eye(cf. Figs. 6a,b). Although there are notable differencedupdraft centers in the eyewall (Fig. 6c), more than 65%of the peak axisymmetric eyewall updraft intensity isquasi-balanced. By comparison, the quasi-balanced vbands in the outer regions vanish after azimuthal av-erage (cf. Figs. 6b and 5b), suggesting that they arehighly asymmetric. Unbalanced axisymmetric v bandsalso show some evidence of propagating gravity waves,especially in the vicinity of the eyewall (Fig. 6c), sug-gesting that the eyewall may be the energy source ofinertial gravity waves.


FIG. 6. As in Fig. 5, but for height–radius cross sections of verticalmotion at intervals of 0.2 m s21 within the radius of 150 km.

FIG. 7. As in Fig. 2, but for height–radius cross sections of radialwinds at intervals of 2.5 m s21 within the radius of 150 km. Solidand dashed lines denote radial outflow and inflow, respectively.

It is apparent from Fig. 7 that the PV inversion systemreproduces reasonably well a general radial inflow inthe lowest 2 km from the outer to inner-core regions,and a radial outflow, which is supergradient (Zhang etal. 2001), from the bottom eye to the upper-level outflow(cf. Figs. 7a,b). As discussed in Zhang et al. (2001), thesupergradient outflows result from the rapid upwardtransport of absolute angular momentum such that thelocal centrifugal force exceeds the radial pressure gra-dient force. The radial outflows tend to spin down a

cyclone’s intensity. However, the inverted radial flowsappear to ‘‘underestimate’’ the simulated systematically(Fig. 7c). In particular, the azimuthally averaged low-level outflow jet associated with the Ekman pumping(Willoughby 1979) is quite weak; it is instead replacedby a deep radial inflow. As discussed in Liu et al. (1999)and Zhang et al. (2001, 2002), this outflow jet plays animportant role in transporting high-ue air from the eyeto support deep convection in the eyewall and in draw-ing air out of the eye to reduce the central pressure. The


FIG. 8. As in Fig. 2, but for potential temperature deviations u at9rintervals of (a), (b) 28C, and at intervals of (c) 18C.

underestimated outflow jet and convergent inflow in thePBL explain why weaker upward motion in the lowest1–2-km layer of the eyewall is obtained (cf. Figs. 5cand 7c). Nonetheless, the inverted and simulated radialflow differences are consistent with the unbalanced ten-dencies obtained by Zhang et al. (2001), so they couldbe treated as unbalanced radial flows. Specifically, theirlocally calculated radial momentum budget shows thepresence of supergradient tendencies (a) in the eyewallfrom the bottom of the eye center to the upper outflowlayer, (b) in association with the low-level outflow jet,(c) with the upper-level return inflow in the eye, and(d) the presence of subgradient tendencies in the lowest1-km airflows outside the RMW (cf. their Figs. 6f and7c herein). Apparently, radial flows exhibit a more sig-nificant portion of unbalanced component than tangen-tial and vertical flows (cf. Figs. 4c, 6c, and 7c), sincethe radial outflows (inflows) represent a cross-isobariccomponent of hurricane flows, pointing toward higher(lower) pressures. Figure 7b shows that about 50% ofradial (cross isobaric) flows could be considered quasibalanced.

Finally, we compare in Fig. 8 the inverted virtualpotential temperature deviations to the simulated. Allu9rdeviations are obtained by subtracting their azimuthallyand radially averaged values at individual heights. Theinverted shows an intense warm core and an intenseu9rinversion near z 5 3 km in the eye, and strong thermalgradients across the sloping eyewall, which are all sim-ilar to the simulated. The inverted is 2–5 K warmeru9rthan the simulated in the lowest 1–2 km of the eyewall,as can also be seen from decreased curvatures (cf.u9rFigs. 8a,b). This warmer boundary layer is attributableto the inverted weaker upward motion causing less adi-abatic cooling and the inverted weaker inflow and out-flow couplet near the top of the PBL (cf. Figs. 5c, 7c,and 8c). Otherwise, the total mass field is extremelywell recovered, indicating that unbalanced mass per-turbations above the PBL are indeed small. The smallmass perturbations appear to be the characteristics ofpropagating gravity waves.

5. Summary and conclusions

In this study, a PV inversion system in pseudoheightZ coordinates has been derived for hurricane flows, fol-lowing DE91, and then tested using an explicit simu-lation of Hurricane Andrew (1992) with the finest gridsize of 6 km (Liu et al. 1997, 1999). This PV inversionsystem also includes the nonconservative effects of fric-tion in the PBL, diabatic heating, and water loading indeep convection. It requires hurricane flows to be stat-ically and inertially stable but allows for the presenceof negative PV, thus reducing the extent of data mod-ification. In order to better linearize the NLB–PV equa-tions, azimuthally averaged (axisymmetric) flows inGWB are defined as a reference state and the remainingportions as perturbations. Since hurricane vortices are

to some extent highly axisymmetric, DE91’s ad hoc lin-earization assumption is abandoned and the so-obtainedperturbation solution can be superposed onto the ref-erence state to recover the balanced nondivergent windsand associated mass fields. To circumvent the high no-nellipticity associated with intense anticyclonic shearvorticity that often occurs outside the RMW and in theupper outflow layers, a small parameter « (50.5) isintroduced into the NLB equation that is then combinedwith the PV equation. This small parameter is used toreduce the influence of anticyclonic contribution while


enhancing the effects of static stability on the inducedcirculations associated with a given q9 distribution. Di-vergent winds are obtained via the quasi-balanced vequation with the inverted balance flow as input, as-suming that the 3D distribution of diabatic heating, PBLeffects, and hurricane vortices are known.

It is shown that the PV inversion system could recoverwell many of the typical features occurring in a hurri-cane, given 3D PV, boundary conditions and all thephysics forcings. They include intense (subgradient)convergent inflow in the PBL and maximum tangentialwinds near its top, sloping RMW located outside theupdraft ridge axis with supergradient flows in the eye-wall, a low-level thermal inversion and a warm-coredeye with intense thermal gradients across the eyewall,organized updrafts in the eyewall and subsidence in theeye, and upper-level anticyclonic-divergent outflow.

However, the PV inversion system appears to under-estimate the bottom cyclonic inflows, the Ekman pump-ing-induced upward motion and thermal perturbationsin the PBL, and anticyclonic outflow in the upper levels.Although the use of coarser vertical resolution in rep-resenting the frictional effects and the required non-negative absolute vorticity may have contributed tosome of the underestimated magnitudes, a significantportion of their differences from the simulated could betreated as unbalanced. Relatively speaking, a more sig-nificant portion (ø50%) of radial flows appears to beunbalanced, since the radial outflows (inflows) representa cross-isobaric component of hurricane flows, pointingtoward higher (lower) pressures. In particular, the low-level outflow jet, containing little quasi-balanced com-ponent, is highly supergradient and unbalanced. Allthese findings are well supported by both locally cal-culated momentum budgets and globally inverted winds.

Despite the presence of more significant unbalancedradial flows, the inverted and simulated mass and winddifferences are found to be generally small above thePBL, particularly after their azimuthal averages. Thus,we may conclude that (a) 3D hurricane flows are to acertain extent quasi-balanced, including the intense ro-tational winds, organized eyewall updrafts and subsi-dence in the eye, cyclonic inflow in the PBL, and upper-level anticyclonic outflow; and (b) the present PV in-version system is capable of recovering almost all typ-ical features in a hurricane, provided that diabaticheating and surface friction together with hurricane vor-tices could be accurately specified.

It should be mentioned that the quasi-balanced flowspresented above are obtained from a simulated hurricanethat is highly axisymmetric. It is desirable to apply thepresent PV inversion system to some highly asymmetricstorms to see to what extent the above conclusion isapplicable. Moreover, in this study we have treated mostof the differenced mass and wind fields as unbalanced,propagating inertial gravity waves without providing de-tailed analysis. In a forthcoming paper, the PV inversionsystem developed herein will be utilized to separate the

unbalanced (wave) from quasi-balanced components ofthe hurricane flows and then study their individual char-acteristics and interactions.

Acknowledgments. We would like to thank Domi-nique Moller, Lloyd Shapiro, Yongsheng Chen, and ananonymous reviewer for their constructive comments,and Mike Montgomery for his helpful discussion. Thiswork was supported by the NSF Grant ATM-9802391,the NASA Grant NAG-57842, and the ONR GrantN00014-96-1-0746.

APPENDIX A

Equations of Atmospheric Motion inz Coordinates

Most of the previous PV diagnostic studies used pas a vertical coordinate (e.g., DE91; Shapiro and Frank-lin 1995; Olsson and Cotton 1997), where p is the Exnerfunction [p 5 Cp(P/p0)k, k 5 Rd/Cp]. For the conve-nience of using the multigrid solver of MUDPACK byAdams (1991), it is desirable to adopt the pressure-dependent, pseudoheight z coordinates of Hoskins andBretherton (1972) that are natural vertical coordinatespointing upward. The variables z and p are related by

z 5 (1 2 p/C )z ,p a (A1)

where za 5 (Cp/Rd)Hs, and Hs 5 p0/r0g 5 Rdu0/g isthe scale height and u0 is the reference potential tem-perature.

The equations of motion for a hydrostatic atmosphere,including the water loading effects, in z coordinates canbe written as

]u ]u ]u ]u ]f1 u 1 y 1 v 2 f y 5 2 1 F ,x]t ]x ]y ]z ]x

]y ]y ]y ]y ]f1 u 1 y 1 v 1 fu 5 2 1 F ,y]t ]x ]y ]z ]y

]f g5 u ,r]z u0

]u ]u ]u ]ur r r r ˙1 u 1 y 1 v 5 Q ,r]t ]x ]y ]z

]u ]y ]2C /R C /Ry d y d1 1 (z 2 z ) [(z 2 z ) v] 5 0, (A2)a a]x ]y ]z

where most of the symbols assume their usual meteo-rological meaning. Because of the importance of waterloading in determining the hydrostatic balance in theeyewall of hurricanes (see Zhang et al. 2000), it is highlydesirable to include the effects of airborne moisture (i.e.,vapor and cloud condensate) and precipitation particles(i.e., rain drop, snow, and hail/graupel) on atmospherictemperature. This is done herein through ur, which isdefined as the virtual potential temperature includingthe water loading effects


R /Cd pp 1 1 q /n0 yu 5 T 5 u , (A3)r r1 2p 1 1 q 1 q 1 qy liquid solid

where y 5 Rd/Ry ø 0.622; and qy , qliquid, and qsolid rep-resent vapor, liquid, and solid cloud particles, respec-tively. The diabatic heat rate Qr in (A2) is given by

1 1 q /n duyQ 5r 1 1 q 1 q 1 q dty liquid solid

d 1 1 q /ny1 u . (A4)1 2dt 1 1 q 1 q 1 qy liquid solid

Based on the scale analysis (not shown) and the resultsof Zhang et al. (2002), the second term on the rhs of(A4) is one order of magnitude smaller than the firstterm, so the former is omitted in our computation ofQr.

APPENDIX B

Nondimensionalization of the PVDiagnostic Equations

To efficiently solve for the balanced mass and windfields, all the variables in the NLB and PV equationsneed to be nondimensionalized. For this purpose, wedefine X 5 Lx, Y 5 Ly, Z 5 Hz, P 5 p0 p, F 5 f 0 f ,where the characteristic parameters L and H have a unitof meter, and p0 5 105 Pa, f 0 5 1024 s21. This givesF 5 L2 f and C 5 L2f 0c, Fx 5 Lf x, and Fy 52 2f f0 0

Lf y; fr 5 ( f x, f y). Substituting the above relations2f 0

into Eqs. (4) and (5) leads to

2 2 2 2 4 2L f L f L f ]c ]c0 0 02¹ f 5 = · ( f =c) 1 2J ,2 2 4 1 2L L L ]x ]y

2f L01 = · f , (B1)rL

2 2 2 21 u L f L f ] f0 0 02Qq9 5 f f 1 ¹ c0 2 2 21 2[r(Z ) G L H ]z

4 3 2 2 2 2L f ] c ] f ] c ] f02 1 . (B2)2 21 2]L H ]x]z ]x]z ]y]z ]y]z

Since r(Z) 5 r0(P/p0) 5 r0p , letting q9 5C /C C /Cy p y p

qp givesC /Cy p

2 3 21 u L f ] f0 0 2Qq 5 ( f 1 ¹ c)2 2[r G H ]z0

2 2 2 2] c ] f ] c ] f2 1 . (B3)1 2]]x]z ]x]z ]y]z ]y]z

Similarly, we can nondimensionalize the hydrostaticequation in (A2) as

2 2L f ]f G0 5 Qu. (B4)H ]z u0

If we assume Q 5 (u0/r0G)(L2 /H 2) and Q 5 L2 u0/3 2f f0 0

GH as the respective characteristic scales of PV andpotential temperature, then Eqs. (B1), (B3), and (B4)will correspond exactly to Eqs. (49), (59), and (12), re-spectively.

REFERENCES

Adams, J., 1993: MUDPACK-2: Multigrid software for approximat-ing elliptic partial differential equations on uniform grids withany resolution. Appl. Math. Comput., 53, 235–249.

Arnason, G., 1958: A convergent method for solving the balanceequation. J. Meteor., 15, 220–225.

Bishop, C. H., and A. J. Thorpe, 1994: Potential vorticity and theelectrostatic analogy: Quasi-geostrophic theory. Quart. J. Roy.Meteor. Soc., 120, 713–731.

Challa, M., and R. L. Pfeffer, 1980: Effects of eddy fluxes of angularmomentum on the model hurricane development. J. Atmos. Sci.,37, 1603–1618.

Davis, C. A., 1992: Piecewise potential vorticity inversion. J. Atmos.Sci., 49, 1397–1411.

——, and K. A. Emanuel, 1991: Potential vorticity diagnostics ofcyclogenesis. Mon. Wea. Rev., 119, 1929–1953.

——, and M. L. Weisman, 1994: Balanced dynamics of mesoscalevortices produced in simulated convective systems. J. Atmos.Sci., 51, 2005–2030.

——, E. D. Grell, and M. A. Shapiro, 1996: The balanced dynamicalnature of a rapidly intensifying oceanic cyclone. Mon. Wea. Rev.,124, 3–26.

Eliassen, A., 1952: Slow thermally or frictionally controlled merid-ional circulation in a circular vortex. Astrophys. Norv., 5, 19–60.

Franklin, J. L., S. J. Lord, S. E. Feuer, and F. D. Marks, 1993: Thekinematic structure of Hurricane Gloria (1985) determined fromnested analyses of dropwindsonde and Doppler radar data. Mon.Wea. Rev., 121, 2433–2451.

Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesismodels: Mathematical formulation and solution. J. Atmos. Sci.,29, 11–37.

——, M. E. McIntyre, and A. W. Robertson, 1985: On the use andsignificance of isentropic potential vorticity maps. Quart. J. Roy.Meteor. Soc., 111, 877–946.

Huo, Z.-H., D.-L. Zhang, and J. R. Gyakum, 1999a: The interactionof potential vorticity anomalies in extratropical cyclogenesis.Part I: Static piecewise inversion. Mon. Wea. Rev., 127, 2546–2561.

——, ——, and ——, 1999b: The interaction of potential vorticityanomalies in extratropical cyclogenesis. Part II: Sensitivity toinitial perturbations. Mon. Wea. Rev., 127, 2563–2575.

Krishnamurti, T. N., 1968: A diagnostic balance model for studies ofweather systems of low and high latitudes, Rossby number lessthan 1. Mon. Wea. Rev., 96, 197–207.

Liu, Y., D.-L. Zhang, and M. K. Yau, 1997: A multiscale numericalstudy of Hurricane Andrew (1992). Part I: Explicit simulationand verification. Mon. Wea. Rev., 125, 3073–3093.

——, ——, and ——, 1999: A multiscale numerical study of Hur-ricane Andrew (1992). Part II: Kinematics and inner-core struc-tures. Mon. Wea. Rev., 127, 2597–2616.

Marks, F. D., R. A. Houze, and J. F. Gamache, 1992: Dual-aircraftinvestigation of the inner core of Hurricane Norbert. Part I: Ki-nematic structure. J. Atmos. Sci., 49, 919–942.

McWilliams, J. C., 1985: A uniformly valid model spanning the re-gimes of geostrophic and isotropic, stratified turbulence: Bal-anced turbulence. J. Atmos. Sci., 42, 1773–1774.

Molinari, J., S. Skubis, D. Vollaro, F. Alsheimer, and H. E. Willough-by, 1998: Potential vorticity analysis of tropical cyclone inten-sification. J. Atmos. Sci., 55, 2632–2644.

Moller, J. D., and S. C. Jones, 1998: Potential vorticity inversion for


tropical cyclones using the asymmetric balance theory. J. Atmos.Sci., 55, 259–282.

——, and M. T. Montgomery, 1999: Vortex Rossby waves and hur-ricane intensification in a barotropic model. J. Atmos. Sci., 56,1674–1687.

——, and L. J. Shapiro, 2002: Balanced contributions to the inten-sification of Hurricane Opal as diagnosed from a GFDL modelforecast. Mon. Wea. Rev., 130, 1866–1881.

Montgomery, M. T., and J. L. Franklin, 1998: An assessment of thebalance approximation in hurricanes. J. Atmos. Sci., 55, 2193–2200.

Olsson, P. Q., and W. R. Cotton, 1997: Balanced and unbalancedcirculations in a primitive equation simulation of a midlatitudeMCC. Part II: Analysis of balance. J. Atmos. Sci., 54, 479–497.

Raymond, D. J., 1992: Nonlinear balance and potential-vorticitythinking at large Rossby number. Quart. J. Roy. Meteor. Soc.,118, 987–1015.

Reasor, P. D., M. T. Montgomery, F. D. Marks Jr., and J. F. Gamache,2000: Low-wavenumber structure and evolution of the hurricaneinner core observed by airborne dual-Doppler radar. Mon. Wea.Rev., 128, 1653–1680.

Reed, R. J., M. T. Stoelinga, and Y.-H. Kuo, 1992: A model-aidedstudy of the origin and evolution of the anomalously high po-tential vorticity in the inner region of a rapidly deepening marinecyclone. Mon. Wea. Rev., 120, 893–913.

Sasaki, Y. K., and J. A. McGinley, 1981: Application of the inequalityconstraint in adjustment of superadiabatic layers. Mon. Wea.Rev., 109, 194–196.

Schubert, W. H., and B. T. Alworth, 1987: Evolution of potentialvorticity in tropical cyclones. Quart. J. Roy. Meteor. Soc., 113,147–162.

——, S. A. Hausman, M. Garcia, K. V. Ooyama, and H.-C. Kuo,2001: Potential vorticity in a moist atmosphere. J. Atmos. Sci.,58, 3148–3157.

Shapiro, L. J., 1996: The motion of Hurricane Gloria: A potentialvorticity diagnosis. Mon. Wea. Rev., 124, 2497–2508.

——, and H. E. Willoughby, 1982: The response of balanced hur-ricanes to local sources of heat and momentum. J. Atmos. Sci.,39, 378–394.

——, and M. T. Montgomery, 1993: A three-dimensional balancetheory for rapidly rotating vortices. J. Atmos. Sci., 50, 3322–3335.

——, and J. L. Franklin, 1995: Potential vorticity in Hurricane Gloria.Mon. Wea. Rev., 123, 1465–1475.

——, and ——, 1999: Potential vorticity asymmetries and tropicalcyclone motion. Mon. Wea. Rev., 127, 124–131.

Sundqvist, H., 1970: Numerical simulation of the development oftropical cyclones with a ten-level model. Tellus, 22, 359–390.

Thorpe, A. J., and C. H. Bishop, 1995: Potential vorticity and theelectrostatic analogy: Ertel–Rossby formulation. Quart. J. Roy.Meteor. Soc., 121, 1477–1495.

Trier, S. B., and C. A. Davis, 2002: Influence of balanced motionson heavy precipitation within a long-lived convectively gener-ated vortex. Mon. Wea. Rev., 130, 877–899.

Willoughby, H. E., 1979: Forced secondary circulations in hurricanes.J. Geophys. Res., 84, 3173–3183.

——, 1990: Gradient balance in tropical cyclones. J. Atmos. Sci., 47,265–274.

Wu, C.-C., and K. A. Emanuel, 1995a: Potential vorticity diagnosticsof hurricane movement. Part I: A case study of Hurricane Bob(1991). Mon. Wea. Rev., 123, 69–92.

——, and ——, 1995b: Potential vorticity diagnostics of hurricanemovement. Part II: Tropical storm Ana (1991) and HurricaneAndrew (1992). Mon. Wea. Rev., 123, 93–109.

Zhang, D.-L., Y. Liu, and M. K. Yau, 2000: A multiscale numericalstudy of Hurricane Andrew (1992). Part III: Dynamically in-duced vertical motion. Mon. Wea. Rev., 128, 3772–3788.

——, ——, and ——, 2001: A multiscale numerical study of Hur-ricane Andrew (1992). Part IV: Unbalanced flows. Mon. Wea.Rev., 129, 92–107.

——, ——, and ——, 2002: A multiscale numerical study of Hur-ricane Andrew (1992). Part V: Inner-core thermodynamics. Mon.Wea. Rev., 130, 2745–2763.

Potential Vorticity Diagnosis of a Simulated Hurricane ...dalin/wang-zhang-pvdiag1-formulation-s03.pdf1JULY 2003 WANG AND ZHANG 1593 q 2003 American Meteorological Society Potential

Documents