1JULY 2003 WANG AND ZHANG 1593 Potential Vorticity Diagnosis of a Simulated Hurricane. Part I: Formulation and Quasi-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 ®nal form 27 January 2003) ABSTRACT Because of the lack of three-dimensional (3D) high-resolution data and the existence of highly nonelliptic ¯ows, 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 nonconservative processes of friction, diabatic heating, and water loading. It requires hurricane ¯ows to be statically and inertially stable but allows for the presence of small negative PV. To facilitate the PV inversion with the nonlinear balance (NLB) equation, hurricane ¯ows are decomposed into an axisymmetric, gradient-balanced reference state and asymmetric perturbations. Meanwhile, the nonellipticity of the NLB equation is circumvented by multiplying a small parameter « and combining it with the PV equation, which effectively reduces the in¯uence of anticyclonic vorticity. A quasi-balanced v equation in pseudoheight coordinates is derived, which includes the effects of friction and diabatic heating as well as differential vorticity advection and the Laplacians of thermal advection by both nondivergent and divergent winds. This quasi-balanced PV±v inversion system is tested with an explicit simulation of Hurricane Andrew (1992) with the ®nest grid size of 6 km. It is shown that (a) the PV±v inversion system could recover almost all typical features in a hurricane, and (b) a sizeable portion of the 3D hurricane ¯ows are quasi-balanced, such as the intense rotational winds, organized eyewall updrafts and subsidence in the eye, cyclonic in¯ow in the boundary layer, and upper-level anticyclonic out¯ow. It is found, however, that the boundary layer cyclonic in¯ow and upper-level anticyclonic out¯ow also contain signi®cant unbalanced components. In particular, a low-level out¯ow jet near the top of the boundary layer is found to be highly unbalanced (and supergradient). These ®ndings are supported by both locally calculated momentum budgets and globally inverted winds. The results indicate that this PV inversion system could be utilized as a tool to separate the unbalanced from quasi-balanced ¯ows for studies of balanced dynamics and propagating inertial gravity waves in hurricane vortices. 1. Introduction al. (1985), the potential vorticity (PV) concept has been frequently used for understanding the three-dimensional Nonlinear balanced (NLB) models have been widely (3D) balanced dynamics of extratropical cyclones (e.g., used in theoretical studies to help understand the fun- Reed et al. 1992; Huo et al. 1999a), tropical cyclones damental dynamics of tropical cyclones. The earliest (e.g., Schubert and Alworth 1987), and convectively work could be traced back to Eliassen (1952) who de- generated midlevel mesovortices (Olsson and Cotton veloped an axisymmetric nonlinear balance model to 1997; Trier and Davis 2002). The PV concept is attrac- investigate how a hurricane vortex evolves under the tive due to its conservative property in the absence of in¯uence of latent heat release and surface friction. Sub- diabatic and frictional processes, and its invertibility sequently, various types of NLB models have been de- principle with which a complete 3D ¯ow could be di- veloped to study the balanced characteristics of hurri- agnosed from a known PV distribution, given a balanced cane vortices (e.g., Sundqvist 1970; Challa and Pfeffer condition and appropriate boundary conditions. How- 1980; Shapiro and Willoughby 1982). Balanced dynam- ever, strong nonlinearity in the PV and NLB equations, ics is of particular interest to many researchers because when applied to mesoscale motion, makes it almost im- it enables one to identify and follow signi®cant ¯ow practical to perform the PV inversion. After applying features (in ``slow manifold'') in space and time. an ad hoc linearization, Davis and Emanuel (1991, here- Since the comprehensive review work of Hoskins et after referred to as DE91; and later re®ned by Davis 1992) were able to develop a PV inversion algorithm Corresponding author address: Dr. Da-Lin Zhang, Department of that allows for decomposition of the 3D PV ®eld in a Meteorology, University of Maryland, College Park, MD 20742. piecewise manner for extratropical cyclones. This ad E-mail: [email protected] hoc linearization appears to be necessary to absorb high- q 2003 American Meteorological Society 1594 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 2 2 order nonlinear terms and facilitate the convergence of whenDl /Dt is evaluated by the advective contribution 2 2 2 iterative calculations because their reference PV, deter- n V t /r , where n is the azimuthal wavenumber. This mined from its climatological (i.e., temporal mean) val- implies that the AB theory may not be very accurate ue, and PV anomalies were at the same order of mag- for high wavenumber asymmetries, even though the nitude (e.g., see Zhang et al. 2002, their Fig. 6). Nev- above scaling may have somewhat overestimated the 2 2 ertheless, their PV inversion algorithm has successfully magnitude ofDl /Dt , according to MoÈller and Mont- been used to diagnose the effects of condensational heat- gomery (1999) and MoÈller and Shapiro (2002). ing (Davis and Weisman 1994; Davis et al. 1996), the Despite the above-mentioned advantage of the AB interaction of different PV anomalies (Huo et al. 1999a), theory, the NLB models or similar balance algorithms the impact of removing upper-level perturbations on the have been widely used, due to their relative simplicities, surface extratropical cyclogenesis (Huo et al. 1999b), to diagnose the balanced ¯ows associated with meso- the in¯uence of upper- and low-level PV anomalies on scale convective systems, including intense hurricane the movement (Wu and Emanuel 1995a,b), and the in- vortices (McWilliams 1985; Zhang et al. 2001). More- tensi®cation of tropical cyclones (Molinari et al. 1998). over, the divergent component of 3D ¯ows may be ob- While DE91's algorithm has been successfully used tained from the balanced winds through the quasi-bal- by Wu and Emanuel (1995a,b) to diagnose hurricane anced v equation (Davis et al. 1996; Olsson and Cotton movement and the storm's in¯uence on its track, the PV 1997). Nevertheless, all of the balance formulations anomalies associated with the hurricane vortex have to mentioned above set PV to be a positive threshold value be excluded in assessing the hurricane's total advective everywhere in order to satisfy an ellipticity requirement ¯ow due partly to the use of a climatological PV dis- for ®nding a solution. In fact, PV is often not positive tribution as a reference state and partly to the sensitivity in the upper out¯ow layer of hurricanes where intense of their inversion results to the choice of the hurricane anticyclonic vorticity is present. In addition, because of center. Motivated by weak asymmetries in the hurricane the lack of high-resolution, high quality winds and ther- core, Shapiro (1996), and Shapiro and Franklin (1999) modynamic data, most of the previous PV diagnostic developed a PV inversion algorithm to study hurricane studies focused on the broad-scale balance character- motion by decomposing the horizontal winds into sym- istics of hurricane vortices, and few studies have been metric (vortex) and asymmetric (environmental) com- performed to investigate the quasi-balanced asymmetric ponents. Thus, the removal of hurricane vortices, as characteristics in the inner core regions of a hurricane. required by Wu and Emanuel (1995a,b), was avoided. In this study, a PV inversion system is developed, Based on the asymmetric balance (AB) theory of Sha- following DE91, but their ad hoc linearization is aban- doned and the ellipticity requirement of the NLB equa- piro and Montgomery (1993), MoÈller and Jones (1998) tion is circumvented. This PV inversion system will then developed a PV inversion algorithm to study the evo- be tested using a high-resolution (Dx 5 6 km) explicit lution of hurricane vortices in a primitive-equation mod- simulation of Hurricane Andrew (1992) with the ®fth- el, and later MoÈller and Shapiro (2002) used it to eval- generation Pennsylvania State University±National uate balanced contributions to the intensi®cation of Hur- Center for Atmospheric Research (PSU±NCAR) non- ricane Opal (1995) in a Geophysical Fluid Dynamics hydrostatic mesoscale model (i.e., MM5). Liu et al. Laboratory (GFDL) model forecast. The AB theory re- (1997, 1999) have shown that the triply nested grid (54/ duces to Eliassen's formulation for purely axisymmetric 1816 km) version of MM5 reproduces reasonably well ¯ow. An advantage of the AB theory is that it provides the track, intensity, as well as the structures of the eye, little restriction on the magnitude of divergence (Mont- eyewall, spiral rainbands, the radius of maximum wind gomery and Franklin 1998; MoÈller and Shapiro 2002), (RMW), and other inner-core features as compared to as compared to the small divergence assumed in the available observations and the results of previous hur- NLB models. However, the validity of the AB theory ricane studies. Their simulation provides a complete dy- 2 depends on a nondimensional parameter, that is, (Dl / namically and thermodynamically consistent dataset for Dt2)/(hj ) K 1, which is the ratio of the azimuthal to us to examine the feasibility and accuracy of this PV inertial frequencies in a symmetric vortex, where Dl/ inversion system in diagnosing the quasi-balanced Dt 5]/]t 1 Vt/r)]/]l,Vt is the mean tangential wind, asymmetric characteristics in the storm's inner-core re- and (r, l) denotes the (radius, azimuth) axes of the gions. cylindrical coordinates, and h 5 ( f 1]Vt/]r 1 Vt/r) The next section shows the derivation of the PV in- is the vertical component of mean absolute vorticity and version system, including the PV, NLB, and v equations j 5 f 1 2Vt/r is the inertia parameter.