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 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± 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± 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-
᭧ 2003 American Meteorological Society 1594 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
2 2 order nonlinear terms and facilitate the convergence of whenD /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 ofD /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 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 (⌬x ϭ 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, (D / namically and thermodynamically consistent dataset for Dt2)/( ) 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 D/ inversion system in diagnosing the quasi-balanced -,Vt is the mean tangential wind, asymmetric characteristics in the storm's inner-core reץ/ץ(t ϩ Vt/rץ/ץDt ϭ and (r, ) denotes the (radius, azimuth) axes of the gions. -r ϩ Vt/r) The next section shows the derivation of the PV inץ/Vtץcylindrical coordinates, and ϭ ( f ϩ is the vertical component of mean absolute vorticity and version system, including the PV, NLB, and equations ϭ f ϩ 2Vt/r is the inertia parameter. A scale analysis in pseudoheight Z coordinates. The associated elliptic by Shapiro and Montgomery (1993) indicates that the conditions will be discussed. Section 3 describes com- validity of the AB theory requires the squared local putational procedures of the inversion algorithm as ap- Rossby number plied to a model-simulated hurricane vortex. Section 4 presents the inverted (axisymmetric and asymmetric) nV222/r R2 ϭ t K 1, results as veri®ed against the model-simulated. Some n balanced and unbalanced characteristics of hurricane 1JULY 2003 WANG AND ZHANG 1595
¯ows will be shown. A summary and concluding re- marks will be given in the ®nal 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), it is appropriate to begin with a more general form of Ertel's PV that takes into account of the moisture and precipitation effects (see Schubert et al. 2001): 1 (١ , (1 ´ Q ϭ
where is the total density consisting of the sum of the FIG. 1. Radial distribution of the wavenumber-0 (solid), wavenum- densities of dry air, airborne moisture (vapor and cloud ber-1 (dashed), wavenumber-2 (dotted), and wavenumber-3 (dot- ϫ V is dashed) components of tangential winds at an altitude of 1 km, that ١ condensate), and precipitation; ϭ 2⍀ ϩ are obtained by averaging 15 model outputs at 4-min intervals during the absolute vorticity vector; and is the virtual po- tential temperature with the water loading effect in- the 1-h period from the 56±57-h simulation ending at 2100 UTC 23 Aug 1992. cluded (see appendix A for its de®nition). Schubert et al. (2001) showed that if a ¯ow is balanced, there exists an invertibility principle with Eq. (1) that could yield the balanced mass and wind ®elds from the spatial dis- is a good approximation to the azimuthally averaged tribution of PV. In the present study, the NLB equation winds above the PBL within a 10% error (Willoughby is used as our balance condition. To derive the NLB 1990; Zhang et al. 2001). Under the hydrostatic and nondivergent assumption, equation, the horizontal wind Vh is decomposed into nondivergent and irrotational components by de®ning a Eq. (1) can be rewritten as streamfunction ⌿, and a velocity potential ⌾; that is, ⌽ץ ⌿ץ ⌽222ץ 1 Q ϭ 0 (F ϩٌ⌿2 ) Ϫ ١⌾ϭV ϩ V , (2) h 2 V ϭ (u, ) ϭϪ١⌿ϫk ϩ ZץXץ ZץXץ Zץ ] h ⌿⌾ r(Z) G where | V | K | V | , and ⌿ and ⌾ satisfy the following ⌽ץ ⌿22ץ ⌾ ⌿ relations, respectively: Ϫ , (5) ZץYץ ZץYץ 22 ] (V ϭ D, (3 ´ ϫ Vhhϭ , ٌ⌾ϭ١ ١ ´ ⌿ϭkٌ where Z ϭ [1 Ϫ (P/P )](R/Cp C /G) represents the ver- where is the vertical component of relative vorticity, 0 p 0 tical pseudoheight coordinate, G is dimensional gravi- and D is the horizontal divergence. Given and D, ⌿ tational acceleration, and r(Z) ϭ (P/P )C /Cp is pseu- and ⌾ can be inverted from (3) with appropriate bound- 0 0 ary conditions. dodensity (Hoskins and Bretherton 1972); see appendix With the above de®nition, the NLB equation relating A for a closed set of dynamical equations in Z coor- the horizontal wind to the mass ®eld can be written as dinates used in this study. Given a PV distribution, Eqs. (4) and (5) are a closed set of equations in ⌽ and ⌿ -⌿ that may be solved iteratively to ®nd a solution. Howץ ⌿ץ 2 hr´ F , (4) ever, several issues need to be considered before such ١ ⌿) ϩ 2J , ϩ F١)´ ⌽ϭhhh١ٌ X Y .a solution can be obtained ץ ץ where the capital letters denote dimensional variables, First, Eqs. (4) and (5) could be highly nonlinear, de-
١h isa2D(X, Y) gradient operator, ⌽ is the geopotential pending on their applications, so a unique solution may height, F is the Coriolis parameter, J is a Jacobian op- not always be obtainable. Thus, it is desirable to line-
erator, and Fr represents the frictional force including arize them by decomposing the total hurricane ¯ow into the effects of the numerical diffusion and the planetary a reference state and a perturbation component, as dem- boundary layer (PBL). Equation (4) is generally valid onstrated by DE91 for extratropical cyclones. Prior to
when the divergent component V⌾ is much weaker than the decomposition, one should note that the balanced the rotational component V⌿. This relation could be solution from Eqs. (4) and (5) requires the ¯ow to be applied to a ¯ow with large curvature, provided that the inertially and statically stable (Hoskins et al. 1985), but Froude number and aspect ratio are small (McWilliams the snapshot model ®elds do not always satisfy the sta- 1985). Note that for an axisymmetric vortex on an F bility requirements. Therefore, to ensure the inertial and plane, Eq. (4) reduces to the gradient wind balance static stability of hurricane ¯ow, we (a) set the vertical (GWB) in the absence of friction. Previous observa- component of absolute vorticity to be slightly positive
tional and modeling budget studies showed that GWB (i.e., Ն 0.1 f ); and (b) adjust the vertical structure 1596 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
2 ,.Z Ն 0.01 N 0/G is always held, where azimuthal mean in the eyewall and outer regions (i.eץ/ץ such that N 2 (ϭ10Ϫ4 sϪ1) is the buoyancy frequency, using the Ј K , and Ј K ), unlike in the previous extra- variational method of Sasaki and McGinley (1981). It tropical cyclogensis studies in which the reference is found that procedure (a) tends to be activated mostly and perturbation states were at the same order of mag- in the upper out¯ow layer, whereas procedure (b) takes nitude when temporal means were used. Moreover, place mostly in the boundary layer and aloft in the eye. the higher the wavenumber, the smaller is the mag- Of importance is that the present procedures do not nitude of perturbation winds. These results are similar require Q Ͼ 0 because of the possible existence of to those shown in Shapiro and Montgomery (1993), negative product of horizontal vorticity and gradient and they also conform to the observational ®ndings [see Eq. (5)]. However, after the above procedures, we of Marks et al. (1992), Franklin et al. (1993), and did ®nd that negative PV becomes weaker in the upper Reasor et al. (2000) that the asymmetric components out¯ow layer and this tends to yield positive PV when are less than 20% of the symmetric component. Thus, it is azimuthally averaged. decomposing the total hurricane ¯ow into the refer- To facilitate the subsequent discussions, it would be ence and perturbation components is consistent with more convenient to rewrite Eqs. (4) and (5) in nondi- the basic assumption involved in linearizing the non- mensional form (see appendix B for more details); that linear equations of (4Ј) and (5Ј), as given below. is, Substituting (6) into Eqs. (4Ј) and (5Ј) gives Јץ 22ץ ץ hhЈϭ f ٌ Јϩٌ 2 yץ (hr´ f ,(4Ј ١ ) ϩ 2J , ϩ ١ f)´ ١ hhh ϭٌ yץ xץ 2 ץ2Ј ץ2Ј ץ 2 ץЈ ץ 22ץ ϩ 2 Ϫ 2 ϩ ץ ץ ץ ץ 22222ץ q ϭ ( f ϩٌ2 ) Ϫϩ, 22 22 yץ xץ yץxץ yץxץ yץ xץh 2 zץyץ zץyץ zץxץ zץxץz ץ[] f ץ f 2ץ2Ј ץЈץ22Јץ (5Ј) ϩ 2 Ϫϩϩx y , (7) yץ xץ yץxץy ץx22ץ[] -where the lower cases denote nondimensionalized var iables, unless mentioned otherwise hereafter. Note that and each term in the above equations has one-to-one cor- Ј ץ ץ ץ2222Јץ respondence to that in Eqs. (4) and (5). qЈϭ( f ϩٌ2 ) ϩٌ2 ЈϪ zץxץ zץxץ zץz22ץBecause of the dominant axisymmetric nature of hur- hh ricanes, we decompose the hurricane ¯ow ®elds into ץ22Ј ץЈ ץ 22 ץ ץ22Јץ axisymmetric and its deviation (asymmetric) parts. In ϪϪϪ zץyץ zץyץ zץyץ zץyץ zץxץ zץxץ doing so, we ®rst de®ne the reference geopotential ЈץЈץЈץЈץ22222Јץ -height as its azimuthal average centered at the min (ϩٌ2 ЈϪ Ϫ , (8 imum surface pressure. The nondimensionalized refer- h 2 zץyץ zץyץ zץxץ zץxץ zץ ence variables andq are then obtained from the GWB -y. Although there are still a few nonץ/fץrelation and Eq. (5Ј), respectively. An advantage of this where  ϭ procedure is that we do not need to iteratively solve linear terms involving ⌸ (Ј, Ј) and ⌸ (Ј, Ј)on Eqs. (4Ј) and (5Ј) for and , givenq . [We have tried the right-hand side (rhs) of the above equations, they to calculate ®rst from the model output and then are one order of magnitude smaller than the linear terms, solved Eq. (4Ј) as Poisson equation for . But the re- based on the results given in Fig. 1. Thus, the nonlin- sulting leads to negative static stability near the top earity of Eqs. (7) and (8) is much weaker than that used of the PBL where the tangential wind is peaked.] All to study extratropical cyclones. Furthermore, it can be the nondimentionalized perturbation variables are then readily shown that all these nonlinear terms will be ab- obtained by sorbed if we follow DE91 by inserting * ϭϩ 0.5Ј and * ϭϩ 0.5Ј into Eqs. (7) and (8). Јϭ Ϫ , Јϭ Ϫ , and Second, if we solve (7) for Ј with Ј obtained from qЈϭq Ϫ q. (6) (8), Eq. (7) is an elliptic equation of Ј such that its associated elliptic condition must be met in order to ®nd Figure 1 shows the radial distribution of azimuth- a solution (Arnason 1958). In the case of a stationary ally averaged tangential winds V t (i.e., wavenumber circular vortex, such a condition, given in cylindrical 0) and their perturbation amplitudesVЈt in wavenum- coordinates, can be written as bers 1±3 at an altitude of 1 km where the tangential winds are nearly maximized. The tangential wave- VץVtt numbers 1±3 winds are obtained after performing az- f ϩ 2 f ϩ 2 Ͼ 0. (9) Rץ R imuthal Fourier decomposition in cylindrical coor- dinates. Obviously, the perturbation wind components The above criterion restricts the magnitude of anticy-
R. Our calculations with theץ/Vtץ are about one order of magnitude smaller than the clonic shear vorticity 1JULY 2003 WANG AND ZHANG 1597 simulated Andrew indicate that a sizeable area outside Ͻ 0]. To circumvent the ellipticity requirement, we mul- the RMW above the PBL is nonelliptic due to the pres- tiply Eq. (7) by a parameter , varying between 0 and
R)) 1, and then add it into Eq. (8) to obtainץ/Vtץ)ence of strong anticyclonic shear [i.e., ( f ϩ 2
2Ј ץ 2 ץ2Ј ץЈ ץ 22 ץ 2 ץ2Ј ץЈ ץ 22 ץ2Ј ץ 2ץ f ϩٌ22ЈϭqЈϪ( f ϩٌ) ϩϩϩϩϪٌ2 Ј z2ץ z hץyץ zץyץ zץyץ zץyץ zץxץ zץxץ zץxץ zץxץ zץz22hhץ ץ22Ј ץЈ ץ 22 ץЈ ץ 22 ץЈץЈץ22Ј ץЈץ22Јץ ϩϩϩٌ2 ЈϪ Ϫ 2 Ϫ 2 ϩ yץx 22ץ yץxץ yץxץ yץx22ץy ץ z [ hץyץ zץyץ zץxץ zץxץ f ץ f ץ 2 ץ2Ј ץЈץ22Јץ Ϫ 2 ϪϪϪx y . (10) [ yץ xץ yץxץ yץxץ yץx22ץ
Јץ22Ј ץЈץ22Ј ץ ץ22Јץ (Since the nonlinear terms ⌸ (Ј, Ј) and ⌸ (Ј, Ј ϩϩϩ zץyץ zץyץ zץxץ zץxץ zץxץ zץyץ in the above equation are much smaller than the linear terms, we may assume the following elliptic condition 2 ץ2Ј ץ2Ј ץ 2 ץЈ ץ 22ץ ,for (10) approximately ϩ 2 Ϫ 2 ϩ 22 22 yץ xץ yץxץ yץxץ yץ xץ ץ 22 ץ ץ 22ץ f ϩ 2ϩ f ϩ 2ϩ 2Ј 2 ץЈץ22Јץ zץy 22ץ zץx22ץ ϩ 2 Ϫ . (11) 22 yץxץy ץ xץ[] 2 2ץ Ϫ 4Ͼ0. (9Ј) y Since perturbation variables are much smaller than theirץxץ 2 2 2 reference values, it is important to realize that the so- z (ϭN ) is equivalent to the mean static lution obtained from (10) and (11) is superposable toץ/ ץ where stability. It is evident from (9Ј) that the value of to the reference stateÐan issue raised by Bishop and be used should be as small as possible, unless the at- Thorpe (1994) and Thorpe and Bishop (1995). mosphere is very stable (e.g., in winter storms). How- Third, solving Eqs. (10) and (11) requires appropriate ever, cannot be too small due to its presence in the lateral, top and bottom boundary conditions. From the coef®cient (i.e., f ) on the lhs of (10) in cases of weak hydrostatic equation static stability. In fact, we found that ϭ0 often leads gץ to the divergence of iterative calculations, when solving , (12) for Ј from (10), at points where the static stability is ϭ z 0ץ relatively weak or qЈ is large. On the other hand, when 2 Neumann boundary conditions for Ј can be speci®ed /Z Ն 0.2 N 0ץ/ץ the vertical adjustment criterion of G is used, any value of between 0 and 1 could result at the bottom (z ϭ zB) and top (z ϭ zT) boundaries, in a convergent solution. For the criterion chosen herein respectively, as follows: 2 Ј gץ Ј gץ Z Ն 0.01 N 0/G), the range of ϭ0.1 ϳץ/ץ ,.i.e) 0.8 would yield a solution without encountering any ϭ ЈB; ϭ ЈT. (13) z ץz 00ץ nonellipticity problem. Thus, in this study a compromise ΗΗzϭzzBTϭz value of ϭ0.5 is adopted to reduce the in¯uence of Dirichlet boundary conditions for Ј and Ј are spec- anticyclonic shear vorticity on the elliptic requirement i®ed along the lateral boundaries, where Ј matches the while enhancing the effects of static stability on the simulated or observed, and the tangential Ј gradient induced circulations associated with a given qЈ distri- matches the normal component of horizontal winds sim- bution. ilar to that used by DE91. That is, Subtracting (8) from (7) leads to 2Јץ 22 hhЈϩ( f ϩٌ) Vh ´ n dlٌ Ј Ͷץ z2ץ ϭϪVh ´ n ϩ , (14) sץ f ץ fץЈץ 2ץ ϭ qЈϪٌ2 Јϩf ٌ2 Јϩ ϩϩx y dl y Ͷץ xץ yץ z2ץhh Ј ץ ץ ץЈץЈ ץ ץ2222222Јץ Ϫٌ2 Јϩ ϩ ϩ where l is a path along the lateral boundaries, s is a z vector tangent to that path, and n a vector normal toץxץ zץyץ zץxץ zץxץ zץxץ zץxץ z2ץ h 1598 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 that path. Note that the domain-averaged net divergence, sociated with the balanced ¯ow is presumably in ``slow for example, resulting from the low-level in¯ow and manifold,'' unlike the divergent ¯ow associated with upper-level out¯ow of hurricanes, must be deducted in gravity waves that tends to radiate away from energy order to satisfy the basic assumptions in deriving Eq. sources. (7); so the second term is included on the rhs of Eq. Secondary circulations in hurricanes may result from (14). Of course, this term can be excluded when a large the balanced ¯ow (i.e., and ) and any dynamical and domain is used, for example, by Wu and Emanuel physical processes (e.g., friction fr or diabatic heating
(1995a,b) and Shapiro (1996). qÇ ), which can be determined by an equation and mass continuity equation, in a manner similar to their quasigeostrophic analog. To show this, we start from b. An equation for a quasi-balanced model the vertical vorticity equation The PV inversion discussed above will yield 3D (non- ץ ץ V ´ Ϫ Ϫ ١ ١´( divergent) balanced mass and horizontal wind ®elds that ϭϪ(V ϩ V z ץt hhץ contain little vertical motion or secondary circulations of hurricanes. However, the secondary circulations, con- f ץ f ץ ץ ץ uץ ץ sisting of convergent in¯ow in the PBL, slantwise ver- ϩϪϩϪy x , (15) yץ xץ zץ xץ zץ yץ tical motion in the eyewall, and divergent out¯ow in the upper troposphere, play an important role in deter- and the thermodynamic equation mining the intensity of hurricane vortices and the de- ץ ץ ץ ץ .velopment of clouds and precipitation in the eyewall ϩ u ϩ ϩ ϭ qÇ , (16) Z ץ yץ xץ tץ Clearly, any balanced theory of hurricanes would be incomplete without including the divergent wind com- ponent. However, including the divergent component to where is vertical velocity (positive implies upward a balanced ¯ow would differ from its conventional bal- motion) in pseudoheight z coordinates (see appendix A). 2 2 z Ϫץtץ/[(Eq. (4Ј] ץance de®nition. In this regard, it would be more appro- Performing (g/ 0)ٌh [Eq. (16)] Ϫ -z, and after some manipulations, we obץ/[(Eq. (15] ץpriate to consider the resulting total 3D ¯ow as being f quasi-balanced, because this divergent component as- tain the following quasi-balanced equation
ץ ץ 22 ץ ץץ ץ ץ 22ץ ץץ ץץ 2ץ ϩ f (z Ϫ z)[(Ϫ z Ϫ z) ] Ϫ f ϩϪf Ϫ 2ٌ zץxץ yץ zץyץ xץ zץ zץyץ yץ zץxץ xץzץ zץ·zΆץ z2ץhaa 3 ץ 2 ץ 2 ץ ץ 222ץץ ץץ 2ץ ץ Ϫ 2 ϪϪ ١ ´ ١] Ϫٌ2 V ´ Ϫ f ϩ f ϭ f [V zץyץtץ yץxץ yץxץ yץx22ץ zץtץ zץz hhhhץ z2ץ zz Ϫ zץ a [] f ץ f 2ץץ f ץ fץץ g 2 yyxx (ϩٌhqÇ Ϫ f ϪϪ ϩ, (17 yץ xץ zץtץ yץ xץzץ 0
where ϭ C /Rd, , and are the total streamfunction properties (Raymond 1992). The last three terms are and geopotential height. The dimensional form of Eq. more physically related and the major sources in de- (17) is similar to that derived by Krishnamurti (1968) termining the magnitude of divergent winds and sec- and DE91 except for the use of z coordinates and in- ondary circulations in hurricanes, which will, in turn, clusions of the velocity potential , and the frictional contribute to the intensities of the ®rst four dynamic and water loading effects. Apparently, the vertical mo- forcing terms. tion of the quasi-balanced model can be determined by Apparently, Eq. (17) is not a fully diagnostic equation the rhs forcing terms of Eq. (17), which, from the left because of the existence of two local tendency terms; t in orderץ/f y ץ t, andץ/f x ץ ,tץ/ץ to right, are the differential vorticity advection and the that is, we must know t could be obtained by eitherץ/ץ Laplacians of thermal advection both by nondivergent to ®nd . The term and divergent winds, the differential deformation or Ja- solving Eq. (15), like in Krishnamurti (1968), or taking cobian term, the  effect, latent heating, and the effects a time derivative of Eqs. (4) and (5), like in DE91; the of friction, respectively. The ®rst four forcing terms are former approach is adopted here. The frictional terms t are obtained directly from the modelץ/f y ץ t andץ/f x ץ more dynamically related and diagnosed from the NLB± of PV equation model; in these four terms, divergent winds output. also play an important role in advecting various ¯ow Because of an additional variable included in Eq. 1JULY 2003 WANG AND ZHANG 1599
(17), which will give the divergent, radial component all the 3D data are transformed from the (x, y, z) to the of hurricane vortices, we need to introduce the conti- cylindrical (r, ,z) coordinates with its origin de®ned nuity equation at the minimum sea level pressure of the storm, follow- ing the procedures described in Liu et al. (1999), and ץ 2 Ϫ ϭϪ(zaaϪ z)[(z Ϫ z) ]. (18) then averaged azimuthally to yield the GWB reference ٌ z state. All the reference and perturbation variables areץ Thus, if the diabatic and frictional contributions are nondimensionalized. With 3D perturbation PV (qЈ) from known (e.g., from a model simulation), Eqs. (15), (17), the model output, Eqs. (10) and (11) are iteratively ,(/ solved, subject to the boundary conditions (13) and (14ץ and (18) form a closed set of equations in , , and t. for the perturbation mass and winds; the results are thenץ In summary, the two sets of equations [i.e., Eqs. (10), added to their reference state. After obtaining the bal- (11) and (15), (17), (18)] plus the reference state de®ne anced total mass and winds, Eqs. (15), (17), and (18) completely a 3D quasi-balanced ¯ow ®eld (both non- are iteratively solved with the diabatic and frictional divergent and divergent) through , , and , and a forcing terms from the model output. In doing so, we quasi-balanced mass ®eld through . From these quan- de®ne P ϭ 980 hPa as the bottom boundary in the tities, all the other relevant balanced properties of the present z coordinates, which puts the eye region (with ¯uid system (e.g., temperature) can be deduced. Since the minimum central pressure of less than 925 hPa) the two sets of equations are in forcing and response essentially above the sea surface. Clearly, in this situ- mode, the piecewise partitioning and inversion could be ation, it is not appropriate to specify ϭ 0 as the bottom in principle conducted to gain insight into the impact boundary condition. Thus, the mass continuity equation of various isolated PV perturbations or different phys- (18) is integrated from the model top downward, fol- ical processes on hurricane ¯ows. lowing Olsson and Cotton (1997), assuming | zϭzr ϭ 0. Finally, all the inverted ®elds are averaged temporally over a 1-h period, and then compared to the hourly 3. Computational procedures averaged model outputs that are similar in magnitude To demonstrate how well the inner-core quasi-bal- and structure to those shown in Liu et al. (1999) and anced mass and winds of a hurricane could be inverted Zhang et al. (2001). from a given PV distribution using the algorithm de- veloped in the preceding section, the model-simulated 4. Quasi-balanced ¯ow Hurricane Andrew (1992) data are output over the two ®ne-mesh (18 and 6 km) domains at 4-min intervals In this section, the quasi-balanced hurricane mass and from the 56±57-h integration, valid at 2000±2100 UTC, wind ®elds are inverted with the PV inversion system 23 August 1992. During this period, the storm has en- developed in section 2, and then veri®ed against the tered its mature stage with a (near steady) maximum model simulated in both the west±east and the azi- surface wind of 68 m sϪ1, although its central pressure muthally averaged radius±height cross sections, in order is still decreasing at a rate of 1 hPa hϪ1 (see Liu et al. to examine its capability in representing the quasi-bal- 1997, their Fig. 2). All the snapshot mass and wind anced, asymmetric and axisymmetric structures of inner- ®elds, as well as the forcing (diabatic and friction) and core ¯ows. Figure 2 compares the simulated PV and in- local tendency terms, are then decoupled from their plane ¯ow vectors to the inverted in west±east vertical MM5 ¯ux forms. cross sections. Pronounced asymmetries of the eyewall To reduce the in¯uence of lateral boundaries on the and its associated ¯ow features are apparent (Fig. 2a). ®nal solution [e.g., see Eq. (14)], the ®nest (6 km) res- For example, the western eyewall is characterized by olution domain is extended in the y direction by inter- more intense upward motion and upper-level out¯ow polating the intermediate (18-km resolution) mesh data, with a deeper secondary circulation than those in the leading to the (x, y) dimensions of 124 ϫ 124 points east. This is consistent with the development of more covering a horizontal area of 738 km ϫ 738 km. The intense deep convection and precipitation in the west model data are then interpolated in the vertical from semicircle (see Liu et al. 1997, their Fig. 5b). Of interest MM5's 23 uneven layers to 34 even z coordinates is the concentration of high PV (over 40 PVU; 1 PVU layers to facilitate the PV inversion using the multigrid ϭ 10Ϫ6 m 2 KkgϪ1 sϪ1) in the eye where little latent solver MUDPACK of Adams (1993). Note that the 34 heating occurs; it coincides with the location of an in- (even) layers could not retain the original high vertical tense thermal inversion layer near z ϭ 2.5 km. This resolution model data in the PBL. As will be seen in feature will have to be studied through PV budget cal- the next section, the reduced vertical resolution tends culations and it will be reported in a future article. Of to misrepresent somewhat the boundary layer frictional further interest is the development of a U-shaped PV effects; see Liu et al. (1997, 1999) for a detailed de- structure above the warm core (i.e., near z ϭ 6 km) that scription of the model con®gurations. appears to result from the cyclonic slantwise upward To decompose the simulated hurricane ¯ow into a advection of the high PV associated with latent heat reference (axisymmetric) state and a perturbation state, release in the eyewall and the slow cyclonic downward 1600 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
advection of low PV in the eye. Negative PV is visible in the upper-level out¯ow where anticyclonic vorticity is large. It is encouraging from Figs. 2b and 2c that the re- calculated PV from the inverted mass and wind ®elds and the inverted in-plane ¯ows compare favorably to the simulated, including the magnitude and distribution of PV, and the storm's secondary circulations. In par- ticular, the peak PV in the eye's inversion layer (and another PV center below), which appears to dominate the hurricane vortex, is well recovered, indicating the great capability of this inversion technique in recovering the hurricane ¯ows. However, some differences in the PBL and the upper out¯ow layer are notable. They could be attributed partly to the use of reduced vertical res- olution (i.e., in representing the frictional effects) and of the required nonnegative absolute vorticity and larger static stability, respectively. Nevertheless, the upper-lev- el anticyclonic out¯ow might be in a transition stage in case symmetric instability (i.e., PV Ͻ 0) is operative. Figure 3 compares the inverted primary circulation of the hurricane to the simulated. Evidently, the PV inversion recovers the basic structures and magnitudes of the hurricane's tangential ¯ows, such as weak ¯ows in the eye, intense asymmetric wind maximums (over 65msϪ1) near the top of the PBL, decreasing winds in the PBL, and the slantwise sloping RMW with large negative vertical shears in the eyewall (cf. Figs. 3a,b). As simulated, the inverted RMW axis is also situated outside of the updraft core. However, the globally in- verted ®elds are generally smoother than the simulated (e.g., wavy isotachs and locally reversed vertical shears near z ϭ 2 km and R ϭ 40 km), as expected. Again, more notable differences appear in the PBL where the dynamic unbalance is more pronounced due partly to the surface friction and partly to the previously men- tioned inaccurate representation of the frictional effects, and in the upper levels where the inverted ¯ows are more constrained by rotational dynamics due to the re- quirement of nonnegative absolute vorticity (Fig. 3c). In addition, slightly stronger tangential ¯ows (Ϯ5m sϪ1) are inverted in the vicinity of the eye±eyewall in- terface where signi®cant downdrafts appear. Despite the stated limitations and some possible inversion and in- FIG. 2. West±east vertical cross sections of PV (contoured), su- terpolation errors, most of these differences could be perposed with storm-relative in-plane ¯ow vectors, from (a) the model considered as the unbalanced component of the hurri- output (every 5 PVU), (b) the inverted or recalculated (every 5 PVU), cane ¯ows, especially for the layers above the PBL 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-min where the differenced ®eld is relatively small. intervals during the 1-h period ending at 2100 UTC 23 Aug 1992. Of signi®cance is that the unbalanced tangential Shadings denote the simulated radar re¯ectivity greater than 15 and winds become much weaker after the azimuthal average 35 dBZ, which represents roughly the distribution of precipitation (see Fig. 4c), suggesting that most of them may be as- with two different intensities. Solid (dashed) lines are for positive sociated with inertial gravity waves as those shown by (negative) values. Note that vertical velocity vectors have been am- pli®ed by a factor of 5. Liu et al. (1999) since the magnitudes of other possible waves (e.g., acoustic) are small. This remains a subject for future study. In particular, the inverted radius±height distribution of tangential ¯ows compares extremely well with the simulated (cf. Figs. 4a,b). The globally inverted unbalanced ¯ows (i.e., Ϯ2±4msϪ1), though differing 1JULY 2003 WANG AND ZHANG 1601
FIG. 3. As in Fig. 2, but for tangential winds contoured at intervals FIG. 4. As in Fig. 3, but for height±radius cross sections of tan- Ϫ1 of (a), (b)5ms ; and (c) contoured at 0, Ϯ2.5, Ϯ5, Ϯ7.5, and Ϯ20 gential winds within the radius of 150 km. Note that (c) isopleths Ϫ1 ms . Solid (dashed) lines denote tangential ¯ows [in (a) and (b)] are given at 0, Ϯ2.5, Ϯ5.0, Ϯ10, and Ϯ15 m sϪ1. or their differenced ¯ows [in (c)] into (out of) the page. in structure, are also similar to the locally derived agra- dial out¯ow jet are also consistent with those (about 16 dient winds above the PBL (cf. Fig. 7 in Zhang et al. msϪ1) shown in Zhang et al. (2001, cf. their Fig. 7b, 2001 and Fig. 4c herein). These results all indicate that and Fig. 4c herein). We speculate that this unbalanced the azimuthally averaged tangential ¯ows above the component would be much weaker if the mass ®eld is PBL are approximately in GWB, conforming to the ob- forced toward the wind ®eld. That is, the reference state servational analysis of Willoughby (1990) and the mo- is calculated ®rst and then Eq. (4Ј) is solved as Poisson mentum budget study of Zhang et al. (2001). The in- equation for . As mentioned earlier, this procedure verted large positive (negative) unbalanced tangential tends to generate statically unstable lapse rates near the winds (about 12 m sϪ1) associated with a low-level ra- top of the PBL. 1602 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
inverted maximum updrafts are about 2.0 and 1.0 m sϪ1 in the respective western and eastern eyewall, as com- pared to the simulated 3.5 and 1.6 m sϪ1 (cf. Figs. 5a,b), indicating that about 60%±70% of peak updrafts in the eyewall are quasi-balanced. The updraft and downdraft bands in the outer regions are also reasonably recovered, albeit with different intensities from the simulated. Thus, we may state that the quasi-balanced equation can recover reasonably the vertical motion ®eld of a hurricane for either diagnostic analyses or model ini- tialization, given 3D latent heating and friction along with its vortex ¯ow. After all, vertical motion in hur- ricanes represents the response of a primitive equation or balanced model to diabatic heating and the PBL ef- fects. Of importance is that the Laplacians of latent heating in the eyewall and other dynamical processes, as shown in Eq. (17), yield general (balanced) subsidence in the eye where little physics forcing occurs. However, the upper-level subsidence appears to be overestimated, as compared to the simulated; this is consistent with the reduced anticylonic out¯ow and the increased return ¯ow into the eye in the inverted ®elds (cf. Figs. 5b and 3c). In contrast, the pronounced downdrafts, originating from the upper-level return ¯ow and coinciding with the evaporative cooling at the eye±eyewall interface, do not appear in the inverted ®eld (cf. Figs. 5a,b). This might be a result of the horizontal resolution used (i.e., ⌬x ϭ 6 km) that is too coarse for the model integration and PV inversion to resolve the interface downdrafts radi- ally. On the other hand, these narrow downdrafts are found to be driven mostly by cold advection in radial in¯ow (see Zhang et al. 2002), so they are not deemed to be possibly inverted from the given weak cooling rates. Of further importance is that propagating gravity waves shown in Liu et al. (1999, see their Fig. 8) are completely absent in Fig. 5b or at any other time (not shown). Instead, these waves appear in the differenced ®eld, in both the inner core and outer regions (Fig. 5c); some bands are 90Њ out of phase with the latent heating. These bands are unbalanced by de®nition and must be consistent with their pertinent unbalanced wind and mass ®elds.
FIG. 5. As in Fig. 2, but for vertical motion at intervals of 0.4 m Figure 6 compares the height±radius cross sections sϪ1. Dark shadings denote latent heating rates greater than 10 and 30 of azimuthally averaged, inverted vertical motion to the KhϪ1, whereas light shadings denote latent cooling rates less than simulated. Again, the equation reproduces a robust Ϫ0.5 and Ϫ3KhϪ1. updraft in the eyewall and general subsidence in the eye (cf. Figs. 6a,b). Although there are notable differenced updraft centers in the eyewall (Fig. 6c), more than 65% The inverted secondary circulations from the quasi- of the peak axisymmetric eyewall updraft intensity is balanced equation (17) are compared to the simulated quasi-balanced. By comparison, the quasi-balanced in Fig. 5, which shows well the development of deep, bands in the outer regions vanish after azimuthal av- boundary-layer-rooted and shallow, ``suspended'' slant- erage (cf. Figs. 6b and 5b), suggesting that they are wise updrafts in the western and eastern eyewall, re- highly asymmetric. Unbalanced axisymmetric bands spectively. Of signi®cance is that both the inverted and also show some evidence of propagating gravity waves, simulated vertical motion bands are almost in phase with especially in the vicinity of the eyewall (Fig. 6c), sug- latent heat release, especially in the eyewall and major gesting that the eyewall may be the energy source of rainbands where more intense latent heating occurs. The inertial gravity waves. 1JULY 2003 WANG AND ZHANG 1603
FIG. 7. As in Fig. 2, but for height±radius cross sections of radial FIG. 6. As in Fig. 5, but for height±radius cross sections of vertical winds at intervals of 2.5 m sϪ1 within the radius of 150 km. Solid motion at intervals of 0.2 m sϪ1 within the radius of 150 km. and dashed lines denote radial out¯ow and in¯ow, respectively.
It is apparent from Fig. 7 that the PV inversion system cyclone's intensity. However, the inverted radial ¯ows reproduces reasonably well a general radial in¯ow in appear to ``underestimate'' the simulated systematically the lowest 2 km from the outer to inner-core regions, (Fig. 7c). In particular, the azimuthally averaged low- and a radial out¯ow, which is supergradient (Zhang et level out¯ow jet associated with the Ekman pumping al. 2001), from the bottom eye to the upper-level out¯ow (Willoughby 1979) is quite weak; it is instead replaced (cf. Figs. 7a,b). As discussed in Zhang et al. (2001), the by a deep radial in¯ow. As discussed in Liu et al. (1999) supergradient out¯ows result from the rapid upward and Zhang et al. (2001, 2002), this out¯ow jet plays an transport of absolute angular momentum such that the important role in transporting high-e air from the eye local centrifugal force exceeds the radial pressure gra- to support deep convection in the eyewall and in draw- dient force. The radial out¯ows tend to spin down a ing air out of the eye to reduce the central pressure. The 1604 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60 underestimated out¯ow jet and convergent in¯ow in the PBL explain why weaker upward motion in the lowest 1±2-km layer of the eyewall is obtained (cf. Figs. 5c and 7c). Nonetheless, the inverted and simulated radial ¯ow differences are consistent with the unbalanced ten- dencies obtained by Zhang et al. (2001), so they could be treated as unbalanced radial ¯ows. Speci®cally, their locally calculated radial momentum budget shows the presence of supergradient tendencies (a) in the eyewall from the bottom of the eye center to the upper out¯ow layer, (b) in association with the low-level out¯ow jet, (c) with the upper-level return in¯ow in the eye, and (d) the presence of subgradient tendencies in the lowest 1-km air¯ows outside the RMW (cf. their Figs. 6f and 7c herein). Apparently, radial ¯ows exhibit a more sig- ni®cant portion of unbalanced component than tangen- tial and vertical ¯ows (cf. Figs. 4c, 6c, and 7c), since the radial out¯ows (in¯ows) represent a cross-isobaric component of hurricane ¯ows, pointing toward higher (lower) pressures. Figure 7b shows that about 50% of radial (cross isobaric) ¯ows could be considered quasi balanced. Finally, we compare in Fig. 8 the inverted virtual potential temperature deviationsЈ to the simulated. All deviations are obtained by subtracting their azimuthally and radially averaged values at individual heights. The invertedЈ shows an intense warm core and an intense inversion near z ϭ 3 km in the eye, and strong thermal gradients across the sloping eyewall, which are all sim- ilar to the simulated. The invertedЈ is 2±5 K warmer than the simulated in the lowest 1±2 km of the eyewall, as can also be seen from decreasedЈ curvatures (cf. Figs. 8a,b). This warmer boundary layer is attributable to the inverted weaker upward motion causing less adi- abatic cooling and the inverted weaker in¯ow and out- ¯ow couplet near the top of the PBL (cf. Figs. 5c, 7c, and 8c). Otherwise, the total mass ®eld is extremely well recovered, indicating that unbalanced mass per- turbations above the PBL are indeed small. The small mass perturbations appear to be the characteristics of propagating gravity waves.
5. Summary and conclusions
FIG. 8. As in Fig. 2, but for potential temperature deviations Ј at In this study, a PV inversion system in pseudoheight intervals of (a), (b) 2ЊC, and at intervals of (c) 1ЊC. Z coordinates has been derived for hurricane ¯ows, fol- lowing DE91, and then tested using an explicit simu- lation of Hurricane Andrew (1992) with the ®nest grid to some extent highly axisymmetric, DE91's ad hoc lin- size of 6 km (Liu et al. 1997, 1999). This PV inversion earization assumption is abandoned and the so-obtained system also includes the nonconservative effects of fric- perturbation solution can be superposed onto the ref- tion in the PBL, diabatic heating, and water loading in erence state to recover the balanced nondivergent winds deep convection. It requires hurricane ¯ows to be stat- and associated mass ®elds. To circumvent the high no- ically and inertially stable but allows for the presence nellipticity associated with intense anticyclonic shear of negative PV, thus reducing the extent of data mod- vorticity that often occurs outside the RMW and in the i®cation. In order to better linearize the NLB±PV equa- upper out¯ow layers, a small parameter (ϭ0.5) is tions, azimuthally averaged (axisymmetric) ¯ows in introduced into the NLB equation that is then combined GWB are de®ned as a reference state and the remaining with the PV equation. This small parameter is used to portions as perturbations. Since hurricane vortices are reduce the in¯uence of anticyclonic contribution while 1JULY 2003 WANG AND ZHANG 1605 enhancing the effects of static stability on the induced unbalanced (wave) from quasi-balanced components of circulations associated with a given qЈ distribution. Di- the hurricane ¯ows and then study their individual char- vergent winds are obtained via the quasi-balanced acteristics and interactions. equation with the inverted balance ¯ow as input, as- suming that the 3D distribution of diabatic heating, PBL Acknowledgments. We would like to thank Domi- effects, and hurricane vortices are known. nique MoÈller, Lloyd Shapiro, Yongsheng Chen, and an It is shown that the PV inversion system could recover anonymous reviewer for their constructive comments, well many of the typical features occurring in a hurri- and Mike Montgomery for his helpful discussion. This cane, given 3D PV, boundary conditions and all the work was supported by the NSF Grant ATM-9802391, physics forcings. They include intense (subgradient) the NASA Grant NAG-57842, and the ONR Grant convergent in¯ow in the PBL and maximum tangential N00014-96-1-0746. winds near its top, sloping RMW located outside the updraft ridge axis with supergradient ¯ows in the eye- APPENDIX A wall, a low-level thermal inversion and a warm-cored eye with intense thermal gradients across the eyewall, Equations of Atmospheric Motion in organized updrafts in the eyewall and subsidence in the z Coordinates eye, and upper-level anticyclonic-divergent out¯ow. Most of the previous PV diagnostic studies used However, the PV inversion system appears to under- as a vertical coordinate (e.g., DE91; Shapiro and Frank- estimate the bottom cyclonic in¯ows, the Ekman pump- lin 1995; Olsson and Cotton 1997), where is the Exner ing-induced upward motion and thermal perturbations function [ ϭ C (P/p ), ϭ R /C ]. For the conve- in the PBL, and anticyclonic out¯ow in the upper levels. p 0 d p nience of using the multigrid solver of MUDPACK by Although the use of coarser vertical resolution in rep- Adams (1991), it is desirable to adopt the pressure- resenting the frictional effects and the required non- dependent, pseudoheight z coordinates of Hoskins and negative absolute vorticity may have contributed to Bretherton (1972) that are natural vertical coordinates some of the underestimated magnitudes, a signi®cant pointing upward. The variables z and are related by portion of their differences from the simulated could be treated as unbalanced. Relatively speaking, a more sig- z ϭ (1 Ϫ /Cpa)z , (A1) ni®cant portion (ഠ50%) of radial ¯ows appears to be where z ϭ (C /R )H , and H ϭ p / g ϭ R /g is unbalanced, since the radial out¯ows (in¯ows) represent a p d s s 0 0 d 0 the scale height and is the reference potential tem- a cross-isobaric component of hurricane ¯ows, pointing 0 perature. toward higher (lower) pressures. In particular, the low- The equations of motion for a hydrostatic atmosphere, level out¯ow jet, containing little quasi-balanced com- including the water loading effects, in z coordinates can ponent, is highly supergradient and unbalanced. All be written as these ®ndings are well supported by both locally cal- ץ uץ uץ uץ uץ .culated momentum budgets and globally inverted winds ϩ u ϩ ϩ Ϫ f ϭϪ ϩF , x xץ zץ yץ xץ tץ Despite the presence of more signi®cant unbalanced radial ¯ows, the inverted and simulated mass and wind ץ ץ ץ ץ ץ differences are found to be generally small above the ϩ u ϩ ϩ ϩ fu ϭϪ ϩF , y yץ zץ yץ xץ tץ ,PBL, particularly after their azimuthal averages. Thus we may conclude that (a) 3D hurricane ¯ows are to a gץ -certain extent quasi-balanced, including the intense ro ϭ , z 0ץ -tational winds, organized eyewall updrafts and subsi dence in the eye, cyclonic in¯ow in the PBL, and upper- Çץ ץ ץץ level anticyclonic out¯ow; and (b) the present PV in- ϩ u ϩ ϩ ϭ Q, zץ yץ xץ tץ -version system is capable of recovering almost all typ ץ ץ uץ ical features in a hurricane, provided that diabatic ϪC /Rd C /Rd heating and surface friction together with hurricane vor- ϩϩ(zaaϪ z)[(z Ϫ z) ] ϭ 0, (A2) zץ yץ xץ tices could be accurately speci®ed. It should be mentioned that the quasi-balanced ¯ows where most of the symbols assume their usual meteo- presented above are obtained from a simulated hurricane rological meaning. Because of the importance of water that is highly axisymmetric. It is desirable to apply the loading in determining the hydrostatic balance in the present PV inversion system to some highly asymmetric eyewall of hurricanes (see Zhang et al. 2000), it is highly storms to see to what extent the above conclusion is desirable to include the effects of airborne moisture (i.e., applicable. Moreover, in this study we have treated most vapor and cloud condensate) and precipitation particles of the differenced mass and wind ®elds as unbalanced, (i.e., rain drop, snow, and hail/graupel) on atmospheric propagating inertial gravity waves without providing de- temperature. This is done herein through , which is tailed analysis. In a forthcoming paper, the PV inversion de®ned as the virtual potential temperature including system developed herein will be utilized to separate the the water loading effects 1606 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
R /C 2 322 2 dp If we assume Q ϭ ( 0/ 0G)(L ff00/H ) and ⌰ϭL 0/ p0 1 ϩ q / ϭ T ϭ , (A3) GH as the respective characteristic scales of PV and p 1 ϩ q ϩ qliquidϩ q solid potential temperature, then Eqs. (B1), (B3), and (B4) where ϭ Rd/R ഠ 0.622; and q , qliquid, and qsolid rep- will correspond exactly to Eqs. (4Ј), (5Ј), and (12), re- resent vapor, liquid, and solid cloud particles, respec- spectively. Ç tively. The diabatic heat rate Q in (A2) is given by 1 q / d Ç ϩ REFERENCES Q ϭ 1 ϩ q ϩ qliquidϩ qdt solid Adams, J., 1993: MUDPACK-2: Multigrid software for approximat- d 1 ϩ q / ing elliptic partial differential equations on uniform grids with ϩ . (A4) any resolution. Appl. Math. Comput., 53, 235±249. dt1 ϩ q ϩ qliquidϩ q solid Arnason, G., 1958: A convergent method for solving the balance equation. J. Meteor., 15, 220±225. Based on the scale analysis (not shown) and the results Bishop, C. H., and A. J. Thorpe, 1994: Potential vorticity and the of Zhang et al. (2002), the second term on the rhs of electrostatic analogy: Quasi-geostrophic theory. Quart. J. Roy. (A4) is one order of magnitude smaller than the ®rst Meteor. Soc., 120, 713±731. term, so the former is omitted in our computation of Challa, M., and R. L. Pfeffer, 1980: Effects of eddy ¯uxes of angular Ç momentum on the model hurricane development. J. Atmos. Sci., Q. 37, 1603±1618. Davis, C. A., 1992: Piecewise potential vorticity inversion. J. Atmos. APPENDIX B Sci., 49, 1397±1411. ÐÐ, and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, 1929±1953. Nondimensionalization of the PV ÐÐ, and M. L. Weisman, 1994: Balanced dynamics of mesoscale Diagnostic Equations vortices produced in simulated convective systems. J. Atmos. Sci., 51, 2005±2030. To ef®ciently solve for the balanced mass and wind ÐÐ, E. D. Grell, and M. A. Shapiro, 1996: The balanced dynamical ®elds, all the variables in the NLB and PV equations nature of a rapidly intensifying oceanic cyclone. Mon. Wea. Rev., need to be nondimensionalized. For this purpose, we 124, 3±26. Eliassen, A., 1952: Slow thermally or frictionally controlled merid- de®ne X ϭ Lx, Y ϭ Ly, Z ϭ Hz, P ϭ p 0 p, F ϭ f 0 f, where the characteristic parameters L and H have a unit ional circulation in a circular vortex. Astrophys. Norv., 5, 19± 5 Ϫ4 Ϫ1 60. of meter, and p 0 ϭ 10 Pa, f 0 ϭ 10 s . This gives Franklin, J. L., S. J. Lord, S. E. Feuer, and F. D. Marks, 1993: The 2 222 ⌽ϭL ff00 and ⌿ϭL f 0,Fx ϭ Lf x, and Fy ϭ kinematic structure of Hurricane Gloria (1985) determined from 2 f 0Lf y; fr ϭ ( f x, f y). Substituting the above relations nested analyses of dropwindsonde and Doppler radar data. Mon. into Eqs. (4) and (5) leads to Wea. Rev., 121, 2433±2451. Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesis ,. models: Mathematical formulation and solution. J. Atmos. Sciץ ץ Lf22 Lf 22 Lf 42 .١) ϩ 02J , 29, 11±37 f)´ ١ 00ٌ2 ϭ y ÐÐ, M. E. McIntyre, and A. W. Robertson, 1985: On the use andץ xץLL22 L 4 signi®cance of isentropic potential vorticity maps. Quart. J. Roy. fL2 .f , (B1) Meteor. Soc., 111, 877±946 ´ ١ ϩ 0 L r Huo, Z.-H., D.-L. Zhang, and J. R. Gyakum, 1999a: The interaction of potential vorticity anomalies in extratropical cyclogenesis. Part I: Static piecewise inversion. Mon. Wea. Rev., 127, 2546± ץ Lf2222 Lf 1 .QqЈϭ 000ffϩٌ2 2561 z ÐÐ, ÐÐ, and ÐÐ, 1999b: The interaction of potential vorticityץr(Z) GLH[0 222 anomalies in extratropical cyclogenesis. Part II: Sensitivity to .2 initial perturbations. Mon. Wea. Rev., 127, 2563±2575 ץ 2 ץ 2 ץ 2 ץLf43 Ϫϩ0 . (B2) Krishnamurti, T. N., 1968: A diagnostic balance model for studies of z ] weather systems of low and high latitudes, Rossby number lessץyץ zץyץ zץxץ zץxץLH22 than 1. Mon. Wea. Rev., 96, 197±207. C/CCp /Cp Since r(Z) ϭ 0(P/p 0) ϭ 0p , letting qЈϭ Liu, Y., D.-L. Zhang, and M. K. Yau, 1997: A multiscale numerical qpC/Cp gives study of Hurricane Andrew (1992). Part I: Explicit simulation and veri®cation. Mon. Wea. Rev., 125, 3073±3093. 23 2 - ÐÐ, ÐÐ, and ÐÐ, 1999: A multiscale numerical study of Hur ץ 00Lf 1 -Qq ϭ ( f ϩٌ2) ricane Andrew (1992). Part II: Kinematics and inner-core struc zץ GH22 0 [ tures. Mon. Wea. Rev., 127, 2597±2616. Marks, F. D., R. A. Houze, and J. F. Gamache, 1992: Dual-aircraft - investigation of the inner core of Hurricane Norbert. Part I: Ki ץ 22 ץ ץ 22ץ Ϫϩ. (B3) .z nematic structure. J. Atmos. Sci., 49, 919±942ץyץ zץyץ zץxץ zץxץ ] McWilliams, J. C., 1985: A uniformly valid model spanning the re- Similarly, we can nondimensionalize the hydrostatic gimes of geostrophic and isotropic, strati®ed turbulence: Bal- equation in (A2) as anced turbulence. J. Atmos. Sci., 42, 1773±1774. Molinari, J., S. Skubis, D. Vollaro, F. Alsheimer, and H. E. Willough- 22 by, 1998: Potential vorticity analysis of tropical cyclone inten- Gץ Lf0 ϭ⌰. (B4) si®cation. J. Atmos. Sci., 55, 2632±2644. z 0 MoÈller, J. D., and S. C. Jones, 1998: Potential vorticity inversion forץ H 1JULY 2003 WANG AND ZHANG 1607
tropical cyclones using the asymmetric balance theory. J. Atmos. ÐÐ, and H. E. Willoughby, 1982: The response of balanced hur- Sci., 55, 259±282. ricanes to local sources of heat and momentum. J. Atmos. Sci., ÐÐ, and M. T. Montgomery, 1999: Vortex Rossby waves and hur- 39, 378±394. ricane intensi®cation in a barotropic model. J. Atmos. Sci., 56, ÐÐ, and M. T. Montgomery, 1993: A three-dimensional balance 1674±1687. theory for rapidly rotating vortices. J. Atmos. Sci., 50, 3322± ÐÐ, and L. J. Shapiro, 2002: Balanced contributions to the inten- 3335. si®cation of Hurricane Opal as diagnosed from a GFDL model ÐÐ, and J. L. Franklin, 1995: Potential vorticity in Hurricane Gloria. forecast. Mon. Wea. Rev., 130, 1866±1881. Mon. Wea. Rev., 123, 1465±1475. Montgomery, M. T., and J. L. Franklin, 1998: An assessment of the ÐÐ, and ÐÐ, 1999: Potential vorticity asymmetries and tropical balance approximation in hurricanes. J. Atmos. Sci., 55, 2193± cyclone motion. Mon. Wea. Rev., 127, 124±131. 2200. Sundqvist, H., 1970: Numerical simulation of the development of Olsson, P. Q., and W. R. Cotton, 1997: Balanced and unbalanced tropical cyclones with a ten-level model. Tellus, 22, 359±390. circulations in a primitive equation simulation of a midlatitude Thorpe, A. J., and C. H. Bishop, 1995: Potential vorticity and the MCC. Part II: Analysis of balance. J. Atmos. Sci., 54, 479±497. electrostatic analogy: Ertel±Rossby formulation. Quart. J. Roy. Raymond, D. J., 1992: Nonlinear balance and potential-vorticity Meteor. Soc., 121, 1477±1495. thinking at large Rossby number. Quart. J. Roy. Meteor. Soc., Trier, S. B., and C. A. Davis, 2002: In¯uence of balanced motions 118, 987±1015. on heavy precipitation within a long-lived convectively gener- Reasor, P. D., M. T. Montgomery, F. D. Marks Jr., and J. F. Gamache, ated vortex. Mon. Wea. Rev., 130, 877±899. 2000: Low-wavenumber structure and evolution of the hurricane Willoughby, H. E., 1979: Forced secondary circulations in hurricanes. inner core observed by airborne dual-Doppler radar. Mon. Wea. J. Geophys. Res., 84, 3173±3183. Rev., 128, 1653±1680. ÐÐ, 1990: Gradient balance in tropical cyclones. J. Atmos. Sci., 47, 265±274. Reed, R. J., M. T. Stoelinga, and Y.-H. Kuo, 1992: A model-aided Wu, C.-C., and K. A. Emanuel, 1995a: Potential vorticity diagnostics study of the origin and evolution of the anomalously high po- of hurricane movement. Part I: A case study of Hurricane Bob tential vorticity in the inner region of a rapidly deepening marine (1991). Mon. Wea. Rev., 123, 69±92. cyclone. Mon. Wea. Rev., 120, 893±913. ÐÐ, and ÐÐ, 1995b: Potential vorticity diagnostics of hurricane Sasaki, Y. K., and J. A. McGinley, 1981: Application of the inequality movement. Part II: Tropical storm Ana (1991) and Hurricane constraint in adjustment of superadiabatic layers. Mon. Wea. Andrew (1992). Mon. Wea. Rev., 123, 93±109. Rev., 109, 194±196. Zhang, D.-L., Y. Liu, and M. K. Yau, 2000: A multiscale numerical Schubert, W. H., and B. T. Alworth, 1987: Evolution of potential study of Hurricane Andrew (1992). Part III: Dynamically in- vorticity in tropical cyclones. Quart. J. Roy. Meteor. Soc., 113, duced vertical motion. Mon. Wea. Rev., 128, 3772±3788. 147±162. ÐÐ, ÐÐ, and ÐÐ, 2001: A multiscale numerical study of Hur- ÐÐ, S. A. Hausman, M. Garcia, K. V. Ooyama, and H.-C. Kuo, ricane Andrew (1992). Part IV: Unbalanced ¯ows. Mon. Wea. 2001: Potential vorticity in a moist atmosphere. J. Atmos. Sci., Rev., 129, 92±107. 58, 3148±3157. ÐÐ, ÐÐ, and ÐÐ, 2002: A multiscale numerical study of Hur- Shapiro, L. J., 1996: The motion of Hurricane Gloria: A potential ricane Andrew (1992). Part V: Inner-core thermodynamics. Mon. vorticity diagnosis. Mon. Wea. Rev., 124, 2497±2508. Wea. Rev., 130, 2745±2763.