A New Look at the Problem of Tropical Cyclones in Vertical Shear Flow: Vortex Resiliency

VOL. 61, NO.1 JOURNAL OF THE ATMOSPHERIC SCIENCES 1JANUARY 2004

PAUL D. REASOR AND MICHAEL T. M ONTGOMERY Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

LEWIS D. GRASSO CIRA/Colorado State University, Fort Collins, Colorado

(Manuscript received 8 April 2003, in ®nal form 26 June 2003)

ABSTRACT A new paradigm for the resiliency of tropical cyclone (TC) vortices in vertical shear ¯ow is presented. To elucidate the basic dynamics, the authors follow previous work and consider initially barotropic vortices on an f plane. It is argued that the diabatically driven secondary circulation of the TC is not directly responsible for maintaining the vertical alignment of the vortex. Rather, an inviscid damping mechanism intrinsic to the dry adiabatic dynamics of the TC vortex suppresses departures from the upright state. Recent work has demonstrated that tilted quasigeostrophic vortices consisting of a core of positive vorticity surrounded by a skirt of lesser positive vorticity align through projection of the tilt asymmetry onto vortex Rossby waves (VRWs) and their subsequent damping (VRW damping). This work is extended here to the ®nite Rossby number (Ro) regime characteristic of real TCs. It is shown that the VRW damping mechanism provides a direct means of reducing the tilt of intense cyclonic vortices (Ro Ͼ 1) in unidirectional vertical shear. Moreover, intense TC-like, but initially barotropic, vortices are shown to be much more resilient to vertical shearing than previously believed. For initially upright, observationally based TC-like vortices in vertical shear, the existence of a ``downshear-left'' tilt equilibrium is demonstrated when the VRW damping is nonnegligible. On the basis of these ®ndings, the axisymmetric component of the diabatically driven secondary circulation is argued to contribute indirectly to vortex resiliency against shear by increasing Ro and enhancing the radial gradient of azimuthal-mean potential vorticity. This, in addition to the reduction of static stability in moist ascent regions, increases the ef®ciency of the VRW damping mechanism.

1. Introduction equate.'' One of the principal questions is: Do the details The impact of environmental vertical shear on the of cumulus convection determine whether a given storm formation and intensi®cation of tropical cyclones (TCs) shears apart or remains vertically aligned, or can the has long been recognized as a largely negative one, effect of convection be meaningfully parameterized and severely inhibiting their formation when above a thresh- still yield a reasonably accurate forecast? Even more old of approximately 12.5±15 m sϪ1 in the 850±200- basic, what is the role of convection in the vertical align- mb layer (Zehr 1992) and weakening or limiting the ment of TCs in shear? development of mature storms (Gray 1968; DeMaria When examining the causes of TC resiliency in ver- 1996; Frank and Ritchie 2001). The error in TC intensity tical shear, we believe the hypothesis that moist pro- prediction using statistical models (DeMaria and Kaplan cesses (e.g., deep cumulus convection) supersede mech- 1999) and operational dynamical models (Kurihara et anisms intrinsic to the dry ``adiabatic'' dynamics is al. 1998) generally increases when vertical shear is a without merit. While it is generally acknowledged that factor. The error of the dynamical models is believed diabatic processes are essential to the maintenance of in part due to de®ciencies in the prediction of vertical TCs in both quiescent and vertically sheared environ- shear in the vicinity of the TC and also from inadequate ments (Ooyama 1969; Shapiro and Willoughby 1982; representation of the interaction of the TC with the ver- Montgomery and Enagonio 1998; Davis and Bosart tical shear ¯ow. Regarding the latter issue, studies of 2001), we will argue here that without the dry adiabatic TCs in vertical shear have sought to de®ne what is ``ad- mechanisms (modi®ed by the presence of convection) TCs would have little chance of surviving differential Corresponding author address: Dr. Paul D. Reasor, Dept. of Me- advection by the vertical shear ¯ow. teorology, The Florida State University, Tallahassee, FL 32306-4520. In a comparison of dry and moist numerical simu- E-mail: [email protected]. lations of an initially weak vortex embedded in weak

᭧ 2004 American Meteorological Society 3 4 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

TC might evolve once vertical shear strong enough to tilt the vortex is imposed. In Fig. 1a the initial undis- turbed TC is shown with cyclonic swirling ¯ow (into and out of the page) and an axisymmetric diabatically driven secondary circulation. When the TC is embedded in westerly environmental vertical shear, the vortex is differentially advected so as to yield a west-to-east tilt with height (Fig. 1b). The asymmetric secondary circulation required to keep the tilted TC quasi-balanced lifts parcels downshear and causes descending motion upshear (Jones 1995). Convection in the TC core will respond asymmetrically, consistent with both observations (e.g., Marks et al. 1992; Reasor et al. 2000; Black et al. 2002; Corbosiero and Molinari 2003; Zehr 2003) and numerical studies (e.g., Wang and Holland 1996; Bender 1997; Frank and Ritchie 1999, 2001). Convec- tion increases downshear and is suppressed upshear at the expense of the axisymmetric component of convection. With the axisymmetric component of the secondary circulation now diminished, the TC becomes more susceptible to differential advection by the vertical shear. Although the secondary circulation and axisymmetrization of convectively generated potential vorticity (PV) anomalies (e.g., Montgomery and Enagonio 1998; MoÈll- er and Montgomery 2000) may continue to intensify the symmetric component of the TC, the question remains how to reduce the tilt asymmetry. Jones (1995; hereafter SJ95) showed that, when a FIG. 1. Schematic of the TC alignment mechanism when the TC barotropic vortex is tilted by vertical shear, the coupling vortex is tilted by vertical shear. (a) The primary TC circulation (into and out of the page) is maintained in the presence of frictional drag between upper- and lower-level cyclonic PV anomalies through stretching of mean TC vorticity by the diabatically driven results in cyclonic precession of the vortex. The pre- axisymmetric secondary circulation. (b) Vertical shear causes the TC cession upshear reduces the vortex tilt from what would to tilt (with vertical tilt angle ␣). Eyewall convection responds by otherwise occur if the vortex were simply advected by becoming increasingly asymmetric, with a maximum in the down- the vertical shear ¯ow. In her benchmark simulation shear quadrant of the storm. The asymmetric component of convection increases at the expense of the symmetric component shown in without moist processes using a hurricane-strength (40 (a). No longer able to offset the frictional drag, the mean TC weakens. msϪ1) vortex embedded in 4 m sϪ1 (10 km)Ϫ1 vertical In the absence of an intrinsic mechanism to realign, the TC will shear shear, the vortex did precess, but the vortex tilt increased apart. (c) Recent work has identi®ed an intrinsic alignment mecha- with time. Contrary to the results of SJ95, in a dry nism called VRW damping. VRW damping counters differential advection of the TC by the vertical shear ¯ow. (d) For suf®ciently strong adiabatic simulation of a hurricane vortex embedded in Ϫ1 VRW damping, a quasi-aligned vortex (␣Ј K ␣) is possible with a approximately 7 m s vertical shear through the depth secondary circulation similar to the axisymmetric con®guration of of the troposphere, Wang and Holland (1996) found that (a). the cyclonic portion of the vortex remained upright. The vortex tilted, realigned, and then achieved a quasi-steady tilt ``downshear-left'' over a 72-h period (see their Fig. vertical shear, Frank and Ritchie (1999) found that the 4). In a subsequent study, Jones (2000a) examined the dry vortex sheared apart, while the intensifying vortex role of large-scale asymmetries generated by the ver- in the model with moist physics remained vertically tically penetrating ¯ow of the tilted vortex. She found aligned. Their results af®rm the idea that a diabatically that the impact of such asymmetries on the vortex tilt driven axisymmetric secondary circulation is needed to is dependent on the radial structure of the initial mean stretch vorticity in the TC core, and thus maintain or vortex. For the benchmark pro®le of SJ95, Jones intensify the vortex against frictional dissipation in the (2000a) showed a case where advection across the vor- boundary layer (e.g., Shapiro and Willoughby 1982). tex core by the large-scale asymmetry appeared to re- One should not draw the conclusion, however, that the align the vortex. For a much broader vortex wind pro®le, mechanisms involved in the evolution of the dry vortex the impact of the large-scale asymmetry on the tilt of become irrelevant once convection is included. the vortex core was reduced, yielding an evolution more To help illustrate the nature of the physical problem, in line with Wang and Holland (1996). The results of consider an initially upright TC in a moist, but quiescent Wang and Holland are supported by the recent numerical environment. Figure 1 illustrates schematically how the study of Frank and Ritchie (2001). In a full-physics 1JANUARY 2004 REASOR ET AL. 5 simulation of a hurricane in 5 m sϪ1 vertical shear environments such as vertically sheared ¯ow, we extend through the depth of the troposphere, Frank and Ritchie here the QG VRW alignment theory to the ®nite Rossby found that the shear had negligible impact on the vortex number regime with vertical shear forcing. We argue intensity and vertical tilt out to almost two days after that the strengthening and structural modi®cation of the the shear was imposed. Based on the above recent nu- mean vortex by the axisymmetric component of the dia- merical simulations, it is clear that there exists a subtle batically driven secondary circulation increase the decay dependence of the TC-in-shear dynamics on radial vor- rate of the quasi mode (i.e., the tilt). Thus, while diabatic tex pro®le. Our aim here is to expose the precise nature processes do not supersede the dry adiabatic mecha- of this dependence. nisms in regards to vortex alignment, we believe they A new theory for the alignment of quasigeostrophic augment the ef®ciency of the VRW damping mechanism (QG) vortices, which separates clearly the mean vortex in reducing the tilt. evolution from the evolution of the tilt asymmetry, has To demonstrate the dependence of the VRW damping been developed by Reasor and Montgomery (2001; mechanism on vortex strength and structure we use a hereafter RM01) and Schecter et al. (2002; hereafter primitive equation (PE) model in its fully nonlinear, SMR). Although from different perspectives, these stud- compressible, nonhydrostatic form, as well as a sim- ies demonstrate the fundamental role of vortex Rossby pli®ed hydrostatic Boussinesq version linearized about waves (VRWs) in the relaxation of tilted vortices to an a circular vortex in gradient balance. Since our primary aligned state. We believe this VRW approach is the key aim here is to explain how the tilt of the cyclonic portion to understanding how a TC maintains its aligned struc- of a TC in shear is reduced, we believe it is suf®cient ture in the presence of vertical shear. to consider initially barotropic cyclonic vortices, as in In RM01 a tilted QG vortex with overlapping upper- SJ95. The impact of baroclinic mean vortex structure and lower-level PV cores was decomposed into an az- on the alignment dynamics is reserved for future study. imuthal mean vortex and small, but ®nite, tilt pertur- In situations where the vertical shear is con®ned pri- bation. The subsequent evolution of the vortex tilt per- marily to the upper troposphere, the downshear advec- turbation was captured by a linear VRW mechanism. tion of the out¯ow anticyclone found in mature storms At small values of the ratio of horizontal vortex scale may need to be taken into consideration (e.g., Wu and to Rossby deformation radius the vortex precesses Emanuel 1993; Flatau et al. 1994; Wang and Holland through the projection of vortex tilt onto a near-discrete 1996; Corbosiero and Molinari 2003). The in¯uence of VRW, or ``quasi mode'' (Rivest and Farrell 1992; Schec- the out¯ow anticyclone is not, in our opinion, well un- ter et al. 2000). By ``discrete'' we mean the VRW is derstood, and we believe it is worthy of future study. characterized by a single phase speed and propagates The outline of the paper is as follows. Section 2 de- azimuthally without change in radial structure [e.g., an ``edge'' wave on a Rankine vortex (Lamb 1932, 230± scribes the three numerical models used in this study: 231)]. Formally, the evolution of the quasi mode may the ®rst model is a linearized PE model designed to take be described as the superposition of continuum modes, advantage of the near-circular symmetry of the problem similar to a disturbance in rectilinear shear ¯ow (Case and our assumption of simple vortex tilts and vertical 1960). But unlike simple sheared disturbances whose shear pro®les. To elucidate the balanced nature of the spectral distribution is broad, the quasi mode is char- tilted vortex evolution we also employ a linearized ver- acterized by a dominant eigenfrequency, and thus be- sion of the ``Asymmetric Balance'' (AB) model (Sha- haves much like a discrete eigenmode of the linearized piro and Montgomery 1993; McWilliams et al. 2003). system. A vortex supporting a quasi mode will precess Finally, in order to broadly identify the alignment re- cyclonically, as depicted in Fig. 1c. The continuum gime boundary, the dry dynamical core of the nonlinear modes that compose the quasi mode destructively in- Colorado State University±Regional Atmospheric Mod- teract, leading to the decay of the quasi mode and hence eling System (CSU±RAMS) model (Pielke et al. 1992) the vortex tilt (Fig. 1d). In SMR the vortex tilt was is utilized. Section 3 examines the free alignment of viewed instead as a discrete VRW interacting with the broadly distributed vortices at ®nite Rossby number. vortex circulation. For monotonically decreasing vor- Section 4 proposes a new heuristic model for under- ticity pro®les the discrete VRW (i.e., the tilt) is invis- standing the linearized dynamics of the TC in shear cidly damped through a resonance with the ambient ¯ow problem. The simple model based on the VRW damping rotation frequency at a critical radius where PV is re- mechanism predicts a downshear-left tilt equilibrium for distributed within a ``cat's eye.'' This means of vortex vortex pro®les characteristic of observed TCs. Fully alignment is termed ``resonant damping.'' The reso- nonlinear numerical simulations are then utilized in sec- nantly damped discrete VRW is mathematically equiv- tion 5 to further our understanding of how TCs maintain alent to the quasi mode. With the vortex in a quasi- their aligned structures in strong shear (Ͼ10 m sϪ1 over aligned state, the convection, and hence the diabatically the vortex depth) in the absence of convection. Both driven secondary circulation, once again approaches idealized and observationally based radial pro®les of axisymmetry. TC tangential velocity are considered. Implications of To examine the resiliency of mature TCs in hostile these results are discussed in section 6. 6 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

2. Numerical models ticity, respectively. Here ␨␷ϭ rϪ1d(r )/dr is the relative vorticity and f is the constant Coriolis parameter. The a. Linear PE model internal gravity wave phase speed cm ϭ NH/m␲, where To develop our basic model of the TC in shear we N is the Brunt±VaÈisaÈlaÈ frequency and is assumed con- employ a Boussinesq PE model on an f plane, bounded stant. The ®rst term on the rhs of each equation rep- vertically by rigid lids. The model is formulated in a resents Rayleigh damping within a sponge ring of the cylindrical coordinate system (r, ␭,z), where z is the form pseudoheight vertical coordinate (Hoskins and Breth- ␲(r Ϫ r ) erton 1972). Consistent with our emphasis on small but ␣(r) ϭ ␣ sin2 1 , r Ͼ r , (4) ®nite amplitude departures from vertical alignment, the o r Ϫ r 1 []21 governing equations are linearized about a circular where r ϭ 2000 km, r ϭ 4000 km, and ␣ is a damping (mean) vortex in gradient and hydrostatic balance. The 1 2 o rate based on the time it takes a gravity wave to pass mean vortex and hydrostatic perturbation (i.e., the tilt) through the sponge ring. The second term on the rhs of are then decomposed into barotropic and internal baro- Eqs. (1)±(3) is a linearized forcing term associated with clinic modes. As even the linear dynamics at ®nite Ross- an imposed vertical shear ¯ow that is assumed weak by number will be shown to exhibit a rich dynamical compared to the basic state swirling ¯ow. The explicit structure, we focus here on the simplest case of an ini- shear forcing terms are given in section 2c. tially barotropic vortex. Consistent with hurricane ob- The system of equations (1)±(3) is discretized radially servations (e.g., Hawkins and Rubsam 1968) and recent using centered second-order ®nite differences on the numerical studies (Rotunno and Emanuel 1987; MoÈller stencil described in Flatau and Stevens (1989) and and Montgomery 2000; RM01; SMR; Schecter and Montgomery and Lu (1997). The system is integrated Montgomery 2003) we employ isothermal (constant po- in time using a fourth-order Runge±Kutta time-stepping tential temperature) boundary conditions on the top and scheme with a time step less than the maximum value bottom of the vortex. Under the Boussinesq approxi- implied by the Courant±Freidrichs±Lewy (CFL) con- mation, the vertical structure of the modes comprising dition. The duration of most simulations examined here the geopotential is then given by cos(m␲z/H), where m is 10␶ , where ␶ ϭ 2␲/(⍀ L) is a circulation period of is the vertical mode number and H is the physical depth e e the mean vortex at r ϭ L, the radius of maximum tan- of the vortex.1 gential wind (RMW). Unless otherwise stated, a 3000- The perturbation is Fourier decomposed in azimuth. km outer radius domain with radial grid spacing ⌬r ϭ The prognostic equations for each vertical and azimuthal 2.5 km is used. mode, referred to here as the equivalent barotropic sys- The mean PV for a barotropic vortex, q ϭ N 2␩,is tem, are prescribed and the associated tangential velocity ®eld is assumed in gradient balance: ␾ˆ mnץץ in uÃ (r, t) Ã 2 ϩ ⍀ mnϪ ␰␷ mn ϩ ␷ d␾ (r f ␷ ϩϭ. (5ץ tץ΂΃ rdr ϭϪ␣(r)uÃ mnϩ F u,s, (1) The perturbation PV amplitude of the initial vortex ,(in asymmetry derived from Eqs. (1)±(3 ץ ϩ in⍀ ␷Ã mn(r, t) ϩ ␩uÃ mnϩ ␾ˆ mn tr 2ץ ΂΃ m␲ 2 ϭϪ␣(r)␷Ã ϩ F , (2) qÃ mnϭ N ␨ˆ mnϪ ␩␾ˆ mn, (6) mn ␷,s ΂΃H in␷Ã is also prescribed. An iterative solution to (6) and the ץ 1 ץ 2 mn ϩ in⍀ ␾ˆ mn(r, t) ϩ c m(ruÃ mn) ϩ linear barotropic balance equation (e.g., Montgomery ,(rrand Franklin 1998ץ[]trץ΂΃

ϭϪ␣(r)␾ˆ mn ϩ F␾,s, (3) 2 1 dd␾ˆ mn n where uÃ , ␷Ã , and ␾ˆ are the radial velocity, tangential r Ϫ ␾ˆ mn mn mn mn rdr dr r2 velocity, and geopotential Fourier amplitudes, respec- ΂΃ tively, for vertical mode m and azimuthal wavenumber 1 dd␺ˆ n22␩ nd⍀ ϭ r␰ mn Ϫ ␺ˆ Ϫ ␺ˆ , (7) n. The azimuthal mean vortex quantities, ⍀ ϭ ␷ /r, ␰ ϭ 2 mn mn f ϩ 2⍀ , and ␩␨ϭ f ϩ , are the angular velocity, rdr΂΃ dr r r dr modi®ed Coriolis parameter, and absolute vertical vor- is sought in order to obtain an initially quasi-balanced perturbation asymmetric wind (de®ned by the nondiv- ergent streamfunction ␺ˆ ) and geopotential ®eld ␾ˆ . 1 Although the isothermal boundary condition formally breaks mn mn down on the top of shallow vortices whose depth is small compared The initial condition employed here to represent a tilted to the tropopause height, we will continue to employ it here as a ®rst vortex therefore projects minimally onto unbalanced approximation to the baroclinic vortex problem. gravity wave motions. Our choice to focus on the bal- 1JANUARY 2004 REASOR ET AL. 7 anced wind ®eld is supported by the PE numerical study ␲z of SJ95 where the thermal and wind ®elds of her sim- us ϭ U cos(␭) cos , ulated tilted vortices in vertical shear were found to ΂΃H evolve in a manner consistent with balanced dynamics. ␲z ␷ ϭϪU sin(␭) cos , (9) s ΂΃H b. Linear AB model where U is the zonal velocity at z ϭ 0. At z ϭ H the The balanced nature of the vortex alignment mech- zonal velocity is ϪU, so the net mean ¯ow difference anism discussed here is clari®ed through use of the over the depth of the vortex is 2U. Assuming the zonal ¯ow is geostrophically balanced, the geopotential as- asymmetric balance (AB) theory (Shapiro and Mont- sociated with this wind ®eld is gomery 1993). The AB theory has been successfully applied in previous idealized studies of both TC track ␲z and intensity change (Montgomery and Kallenbach ␾s ϭϪfUr sin(␭) cos . (10) 1997; Montgomery et al. 1999; MoÈller and Montgomery ΂΃H 1999, 2000). In the context of vortices supporting quasi The shear ¯ow will be viewed here as a small pertur- modes, Schecter and Montgomery (2003) found that the bation to the mean vortex. Consequently, in the line- AB model yields better agreement with the PE model arization of the primitive equations, products of shear when derived following the ``PV path'' of McWilliams terms and products of shear and vortex perturbation et al. (2003) than when derived following the ``conti- terms are neglected. For m ϭ 1 and n ϭ 1 the shear nuity path'' of Shapiro and Montgomery (1993). We forcing terms in Eqs. (1)±(3) are then as follows: follow the former approach here and also ®nd improved 11d␷ agreement with the PE model throughout the vortex Fu,s ϭ iU⍀, F␷,s ϭϪ U , alignment phase space characteristic of TC-like vortices. 22dr Following the derivation described above for the PE 1 F ϭ fU␷ . (11) system, a prognostic equation for the perturbation geo- ␾,s 2 potential can be derived for the linear equivalent barotropic AB model: Since VRWs are an essential element of the vortex alignment theory of RM01 and its extension here to the ␾ˆ n2 1 forced problem, it proves useful later to express theץ r ץ ␩␰ ץ ϩ in⍀Ϫmn S(r) ϩ ␾ˆ 22 mn shear forcing in terms of its effect on the evolution of rrl[]r (r) · perturbation PV. The linearized PV equation for the PEץ r΂΃␩␰ץ trΆץ΂΃ in␰ d␩ system with vertical shear forcing is AB Ϫ ␾ˆ mn ϭ F ␾,s , (8) dq ץץ r␩ dr ϩ⍀ qmm(r, ␭, t) ϩ u ␭ drץ tץwhere q ϭ ␩␰N 2, as in the PE system, S(r) ϭ (2 Ϫ ΂΃ ␩␩)/ (ϭ1 for the continuity path), andl2 (r) ϭ (NH/ r dq ␷ q m␲) 2()␩␰ Ϫ1 is the square of the local internal Rossby ϭϪU ϩ cos␭, (12) 2 deformation radius (Reasor 2000). The termAB on the Ά·dr lr,G f F ␾,s rhs represents a vertical shear forcing. In the limit of or for n ϭ 1, in®nitesimally small Rossby number, Ro ϭ Vmax/ fL, dq ץ where V is the maximum tangential velocity of the max ϩ i⍀ qÃ (r, t) ϩ uÃ tdrm1 m1ץvortex, we note that Eq. (8) reduces to the equivalent ΂΃ barotropic QG PV equation used by RM01. The primary difference between the QG and AB prognostic equations 1 dq ␷ q ϭϪ U ϩϵF . (13) is the replacement of the ``global'' deformation radius, 2 q,s 2 Ά·dr lr,G f lr,G ϭ NH/m␲ f, with a radially dependent local one, l (r). The shear forcing Fq,s in the PV equation (13) is related r AB to the shear forcing in the AB equation (8) by F ␾,s ϭ 2 ␰Fq,s/N . The ®rst term on the rhs of Eq. (13) is the c. Linearized vertical shear forcing differential advection of the vortex PV by the vertical shear ¯ow. The second term on the rhs produces effec- A unidirectional vertically sheared zonal ¯ow is in- tive ␤ gyres resulting from the advection of zonal PV cluded in the linear PE model in section 5 as a time- (associated with the zonal vertical shear) by the mean invariant forcing with vertical structure of the m ϭ 1 vortex tangential wind. For all of the sheared vortex baroclinic mode. In cylindrical coordinates the radial simulations presented here this latter forcing is found and azimuthal components of the zonal shear ¯ow us to be negligible, approximately one to two orders of are expressed as magnitude less than the ®rst forcing term. According to 8 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

Vandermeirsh et al. (2002) in a study of the splitting, TABLE 1. Summary of primary numerical experiments examining or shearing apart, of oceanic-like QG vortices by vertical the free alignment and alignment in vertical shear of initially barotropic cyclonic vortices. shear ¯ow in a 2½-layer model, the effective ␤ gyres cause the vortex to remain more resilient. The impact Ambient of the ␤ gyres increases with decreasing l , thus in- vertical shear r,G Ϫ1 Vmax (m s / creasing the shear threshold required to split the vortex. Vortex (m sϪ1) L (km) l (km) Ro ␶ (h) 10 km) In this limit in the fully nonlinear context, where the r,G e mean vortex is allowed to evolve, it is actually possible Gaussian 5 200 100±ϱ 0.25 69.8 0 Gaussian 40 100 154 3.9 4.4 0 for the vortex to disperse on the background PV ®eld Gaussian 40 100 630 6.5 4.4 0 as Rossby waves before shearing apart. Again, the TC Gaussian 10 100 1230 3.2 17.5 4 cases considered here are typi®ed by relatively large lr,G Gaussian 40 100 1230 12.7 4.4 4,8 compared to horizontal vortex scale, and hence the in- SJ95 40 100 1230 12.7 4.4 0,4 ¯uence of the effective ␤ gyres is minimal. Olivia 52 17 862 68 0.6 0,10 d. Fully nonlinear, compressible, nonhydrostatic Gaussian vortex in this regime, however, occurs pri- RAMS model marily through axisymmetrization by sheared VRWs. To investigate the modi®cation of the linear results The ®ndings of SMR suggest that the alignment dy- by nonlinear processes we use the dry dynamical core namics depends subtly on the radial vortex structure. of the Regional Atmospheric Modeling System version With this caveat in mind, we will continue to use the 4.29Ðdeveloped at Colorado State University (Pielke Gaussian vortex to present the basic alignment theory et al. 1992). Details of the model and the system of at ®nite Rossby number. In section 5 the theory is ap- equations solved are presented in the appendix. The plied to both non-Gaussian analytic and observationally model domain is 8000 km on a horizontal side (grid based radial vortex pro®les. spacing ⌬x ϭ⌬y ϭ 10 km) and 10 km in the vertical To mimic the simple tilt of a barotropic vortex we (grid spacing ⌬z ϭ 1 km). follow RM01 and add to the mean PV a perturbation PV of the form

3. Vortex alignment at ®nite Rossby number ␲z i␭ qЈ(r, ␭, z) ϭ Re qÃ 11(r)e cos , (15) a. Initial conditions []΂΃H For the experiments presented in this section the ra- where Re[ ] denotes the real part. The amplitude of the dial structure of the azimuthal-mean vortex PV takes PV asymmetry representing the vortex tilt is given by the Gaussian form dq qÃ 11(r) ϭ ␣Ä , (16) Ϫ(␴r)2 q(r) ϭ qemax , (14) dr whereÄ is a constant. We de®ne Ä qq/(d /dr) , where q max is the maximum mean PV and ␴ is the in- ␣ ␣ ϭ ␣ max max verse decay length of the PV pro®le. This pro®le was where (dq/dr)max is the maximum mean PV gradient and used by RM01 as an approximation to the broad vor- ␣ is a nondimensional amplitude. ticity distributions observed in weak TCs (e.g., Wil- In the free-alignment experiments we let ␣ ϭ 0.3. For loughby 1990). Strong TCs are often characterized by this value of tilt amplitude RM01 found good agreement relatively sharp radial gradients of vorticity in the eye- between linear and nonlinear QG simulations for all lr,G. wall region (Marks et al. 1992; Reasor et al. 2000; Kos- In the experiments with zonal vertical shear the initial sin and Eastin 2001), so the Gaussian structure em- vortex is vertically aligned (␣ ϭ 0) and allowed to de- ployed here is a less ideal approximation at large Rossby velop a tilt through the differential advection of vortex number. Still, mature hurricanes are far from Rankine PV by the shear ¯ow. Pertinent details of the primary vortices, with tangential winds outside the vortex core numerical experiments are summarized in Table 1. typically decaying like rϪ0.4 to rϪ0.6 (Riehl 1963; Wil- loughby 1990; Pearce 1993; Shapiro and Montgomery b. Validation of prior QG results 1993), as opposed to rϪ1 for the Rankine vortex. The importance of radial vortex structure in the ver- RM01 showed that, for a broadly distributed QG vor- tical alignment of QG vortices was highlighted by SMR. tex exhibiting small but ®nite amplitude tilt, the evo- For both the Gaussian and ``Rankine-with-skirt'' vor- lution of the vortex is characterized primarily by the tices they found that the tilt decay at L/lr,G Ͻ 2, where ratio, L/lr,G. Using the linear AB theory [see Eq. (8)] as here L is taken to be the RMW, is explained by the a guide to extending the QG results into the ®nite Ross- decay of a quasi mode. For L/lr,G Ͼ 2 the tilted Rankine- by number regime, the appropriate ratio is found to be with-skirt vortex still evolves in a manner consistent L/lr ϭ (L/lr,G)(͙␩␰ / f ). As the Rossby number is in- with a decaying quasi mode. The alignment of the creased from the in®nitesimal values of QG theory, 1JANUARY 2004 REASOR ET AL. 9

FIG. 2. Vertical alignment of an initially tilted Gaussian vortex.

PV intercentroid separation between z ϭ 0 and 10 km after t ϭ 5␶e as a function of the ratio of RMW to global internal Rossby defor- FIG. 3. Azimuthal wavenumber-1 geopotential amplitude | ␾ˆ | mation radius L/lr,G. Here, ␶e is a circulation period of the mean vortex 11 at the RMW. The PE results include ®nite Rossby number effects (normalized by the initial value) at the radius of maximum mean PV neglected in the QG formulation. Details of this simulation with L gradient of an initially tilted Gaussian vortex after t ϭ 5␶e,asa ϭ 200 km are listed in Table 1. function of both Rossby number, Ro, and L/lr,G. This quantity will henceforth be used as our canonical measure of vortex tilt. Small contour values indicate near-complete alignment. Vortices clearly supporting a quasi mode are denoted by ``Q,'' while vortices aligning L/lr also increases from its QG value. Schecter and Montgomery (2003) con®rmed the hypothesis of RM01 exclusively via sheared VRWs are denoted by ``S.'' that an increase in L/lr results in an increased rate of alignment, qualitatively similar to the increase in align- simple vortex tilts, like those described here, they also ment rate achieved by increasing L/lr,G in the QG theory. found a rapid decay of the vortex tilt over the ®rst ␶e Figure 2 illustrates the modi®cation of the QG results not captured by resonant damping theory. Further details of RM01 by ®nite Rossby number effects. Shown is the of this initial vortex behavior and a complete exami- horizontal separation of upper- and lower-level PV cen- nation of the linear alignment phase space typical of troids after t 5 [where 2 /(L)] as a function ϭ ␶e ␶e ϭ ␲ ⍀ TCs are discussed below. of L/lr,G from the linear QG simulations of RM01 and the corresponding linear PE simulations. The vortex in Ϫ1 this case has L ϭ 200 km and Vmax ϭ 5ms , yielding c. Alignment of strong vortices Ro ϭ 0.25. The range of deformation radii was obtained 1) THE LINEAR ALIGNMENT PHASE SPACE by varying the vortex depth. While the inclusion of ®nite Rossby number effects increases the rate of alignment For typical tropical conditions and the gravest vertical in the transition regime between precession and com- modes the global internal deformation radius, lr,G is on plete alignment, the two curves are still in good agree- the order of 1000 km. For horizontal vortex scales on ment. At small values of L/lr,G the vortex precesses while the order of 100 km, L/lr,G ϳ 0.1, well within the quasi slowly aligning through projection of the tilt onto a mode regime according to Fig. 2. If the alignment dy- decaying quasi mode. For L/lr,G Ͼ 2 the vortex aligns namics at ®nite Rossby number depends on the local completely over the 5␶e time period through projection deformation radius in a manner similar to the way the of the tilt onto sheared VRWs. The linear alignment QG system depends on lr,G, then we might anticipate a theory of RM01 therefore remains valid at small but transition from alignment via a decaying quasi mode to ®nite Rossby number. alignment via sheared VRWs as the Rossby number is

For the range of L/lr,G indicated in Fig. 2, SMR dem- increased. We test this hypothesis by mapping out the onstrated that the QG decay rate of the vortex tilt is alignment phase space for a Gaussian vortex of varying predicted by the theory of resonant damping. An inter- Rossby number and typical tropical values of L/lr,G. The esting question is whether the resonant damping theory results using the linearized PE model (1)±(3) are shown can accurately predict the kind of increases in alignment in Fig. 3. The geopotential amplitude, |␾ˆ 11 | , evaluated rate shown in Fig. 2 when the Rossby number is in- at the radius of maximum mean PV gradient is used as creased. A detailed investigation of the resonant damp- a proxy for the tilt of the vortex core. Plotted are values ing theory as it applies to vortex alignment at ®nite of the tilt after t ϭ 5␶e, normalized by the initial vortex Rossby number is presented by Schecter and Montgom- tilt. The evaluation time is somewhat arbitrary, but given ery (2003). For the range of L/lr,G considered here and that the environment of a typical TC changes on even at order unity Rossby number, they found that Gaussian shorter time scales, we deemed this time to be meteo- vortices do indeed support decaying quasi modes. For rologically relevant. For each case shown in Fig. 3 we 10 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 have veri®ed the balanced nature of the vortex evolution to the linear PE simulated result. For pure spiral wind- by comparing the results with the AB theory [Eq. (8)] up behavior the pseudo-PV perturbation evolves as prediction (not shown). The structure of the phase di- qÃ (r, t) ϭ qÃ (r, 0)eϪi⍀(r)t, (18) agram is reproduced. 11 11

It is clear from Fig. 3 that rapid alignment is best where qÃ 11(r, 0) is the initial pseudo-PV amplitude. The achieved by simultaneously increasing the Rossby num- geopotential amplitude |␾ˆ 11 | is then obtained by in- ber and L/lr,G. The upper-right portion of the phase verting Eq. (17) at each time. In the region of the phase space, where the rate of alignment is greatest, is pop- space denoted by ``S'' in Fig. 3 the tilt asymmetry evo- ulated by intense, horizontally large but shallow vor- lution is approximated well by the AB spiral wind-up tices. Conversely, in the lower-left portion of the phase solution (18) out to at least t ϭ 10␶e. On this basis we space, where the initial vortex tilt is maintained, vortices conclude that Gaussian vortices in this regime do not are characteristically weak and horizontally small but support long-lived quasi modes. The physical mecha- deep. These results are in qualitative agreement with the nism for alignment in this region of the phase space is sensitivity study of SJ95. She demonstrated that changes the redistribution of PV by sheared VRWs. To illustrate in the vortex and environmental parameters that increase the vortex evolution in the quasi mode and spiral wind- the penetration depth, and thus yield stronger vertical up regimes we show results from two representative coupling between upper- and lower-level PV anomalies simulations. of the vortex, result in greater resistance of the vortex to vertical tilting by an imposed vertical shear ¯ow. 2) ALIGNMENT BY SPIRAL WIND-UP To determine where in the alignment phase space the quasi mode is relevant to the vortex dynamics, we have In the ®rst experiment we examine the self alignment Ϫ1 examined the solution in each case for evidence of a of a vortex with L ϭ 100 km, Vmax ϭ 40 m s , and quasi mode. The presence of a quasi mode causes the lr,G ϭ 154 km. According to Fig. 2, QG theory at L/lr,G PV asymmetry associated with the vortex tilt to exhibit ϭ 0.65 would predict precession of the vortex with little a discretelike structure, propagating around the mean decay of the tilt over several ␶e. The actual evolution vortex with little change in radial structure in time of the vortex shown in Fig. 4 with Ro ϭ 3.9 is dra- (RM01; SMR; Schecter and Montgomery 2003). For matically different. In Fig. 4a we see that the asymmetric the ®nite Rossby number Gaussian vortex, Schecter and PV at z ϭ 0 undergoes a spiral wind-up similar to the

Montgomery (2003) proved that the quasi mode must small lr,G results of RM01 (see their Fig. 10). It should decay in time and that the rate of decay is exponential, be made clear, however, that the evolution of asymmetric as anticipated by the QG theory of SMR. Cases in which PV is not explained simply by passive advection. Ra- the tilt evolution displays an exponential decay with dius±time plots of the PV amplitude (not shown) in- time are denoted by ``Q'' in Fig. 3. Consistent with the dicate the outward propagation and stagnation of wave QG results of Fig. 2, the quasi mode is present for the packets. Since gravity waves have minimal re¯ection in chosen range of L/lr,G at small Rossby number. When the PV core and the internal gravity wave speed cm ϭ Ϫ1 L/lr,G is small compared to unity, the quasi mode persists 16ms is much greater than the approximately 4 m for Ro k 1. sϪ1 observed radial group velocity, the outward prop-

As L/lr,G increases, a demarcation in solution with agating features must be VRWs. One might reasonably increasing Rossby number becomes evident. The vortex conclude then that the mechanism for the alignment of ceases to re¯ect the presence of a quasi mode. Since the vortex in this part of the phase space is the same the perturbation radial velocity vanishes in this limit of VRW process identi®ed by RM01 in the QG context at small local Rossby deformation radius (see RM01 for large L/lr,G. details), the radial advection of mean PV in the line- Figure 4b further supports this VRW interpretation. arized PV equation [see Eq. (13)] can be neglected com- Shown is the vortex tilt as a function of time out to t pared to the azimuthal advection term (i.e., the advection ϭ 10␶e. Consistent with the ®ndings of Montgomery of perturbation PV by the mean tangential wind). In this and Kallenbach (1997, and references therein), as the limit perturbation PV is then passively advected around perturbation PV disperses on the mean vortex as sheared the vortex. RM01 referred to this evolution in the QG VRWs, the perturbation geopotential decreases in am- context as the ``spiral wind-up'' solution (Bassom and plitude. This decrease occurs rapidly, with the geopo- Gilbert 1998). tential approaching approximately 20% of its initial val-

Because the PE perturbation PV evolution is largely ue in less than 2␶e. The balanced nature of this rapid balanced, we employ the expression for AB pseudo-PV, alignment process is evidenced by the good agreement between PE and AB simulations. A pure spiral wind- ␾ˆ N (2␰ Ϫ ␩)1 up solution derived from Eqs. (17)±(18) is also plottedץ r ץN 22 11 Ϫϩ␾ˆ ϭ qÃ , (17) r ␰␩rl22(r) 11 11 in Fig. 4b. As already indicated in Fig. 3, the simulatedץ r ␰ץ r ΂΃ []r decay of vortex tilt follows the spiral wind-up solution. to diagnose the ®nite Rossby number tilt evolution The oscillations in amplitude, which are present in both

| ␾ˆ 11 | in the spiral wind-up limit, and then compare it the PE and AB models, result from the radial propa- 1JANUARY 2004 REASOR ET AL. 11

FIG. 4. Gaussian vortex alignment at Ro ϭ 3.9 in a regime where QG theory predicts

precession (L/lr,G ϭ 0.64). (a) Evolution of the azimuthal wavenumber-1 component of Bous- sinesq PV at z ϭ 0 (contour interval 1 ϫ 10Ϫ8 sϪ3) showing axisymmetrization in the vortex

core; and (b) log-linear plot of vortex tilt (normalized by the initial value) over a 10␶e period from the linear PE model (solid), linear AB model (dashed), and spiral wind-up solution (dotted). See text for details on the spiral wind-up solution. gation of VRWs past the radius of maximum mean PV of the quasi mode is proportional to the radial gradient gradient used to compute the vortex tilt. of mean PV at the critical radius. Figure 5b con®rms the presence of the quasi mode because the tilt asymmetry at long times decays slowly at an exponential 3) ALIGNMENT BY QUASI-MODE DECAY rate.2 In the second experiment the same mean vortex is Although the quasi mode ultimately dominates the used, but with Ro ϭ 6.5. More signi®cantly lr,G ϭ 630 solution, over the ®rst ␶e the vortex tilt decays more km, yielding L/lr,G ϭ 0.16. The PV evolution shown in like the spiral wind-up evolution of the previous ex- Fig. 5a strongly suggests the presence of an underlying periment. This ``adjustment'' of the vortex at early times discrete VRW structure. Although wavelets are ob- is found to be characteristic of all simulations in the served to propagate radially outward, as in the previous quasi mode regime of the alignment phase space. The case, the PV asymmetry as a whole shows a distinct adjustment becomes more pronounced as the Rossby cyclonic azimuthal propagation with a distinct azi- number is increased. It is not an adjustment accom- muthal phase speed of approximately one-®fth Vmax. The plished by gravity wave radiation since the PE and AB radius at which the phase speed equals the rotation speed of the vortex, the critical radius, falls at r ϭ 250 km. This location is outside the vortex core in a region of 2 The quantitative discrepancy between the PE and AB decay rates is a consequence of the large (Ͼ1) values of the AB expansion pa- near-zero mean radial PV gradient. According to the 2 2 ␭) /␩␰ , outside the vortex core for aץ/ ⍀ץt ϩץ/ץ) rameter, DI ϵ ®nite Rossby number vortex alignment theory of Schec- Gaussian vortex supporting a quasi mode. See Schecter and Mont- ter and Montgomery (2003), the exponential decay rate gomery (2003) for further discussion.

FIG. 5. As in Fig. 4 but for Ro ϭ 6.5 and L/lr,G ϭ 0.16. The initial rapid decay of vortex

tilt over the ®rst ␶e is a consequence of the tilt asymmetry projecting strongly onto sheared VRWs. At later times the tilt decays exponentially, consistent with a resonantly damped VRW. 12 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61 models agree in its timing. The adjustment results from dqЈ ϩ (i␻ Ϫ ␥)qЈϭ0. (20) the ``mismatch'' between the simple tilt initial condition dt p [Eq. (16)] and the quasi mode radial structure. In the QG theory, the two were approximately identical This equation can be motivated physically from a re- (RM01; SMR). Schecter and Montgomery (2003) have casting of the linearized PE PV equation or, since the shown that at ®nite Rossby number the quasi mode no perturbation evolution is largely balanced, the AB pseu- longer takes the form of a simple, spatially uniform tilt. do-PV equation. Consistent with the balanced VRW dy- The general initial tilt perturbation then evolves toward namics, Eq. (20) describes a single-component damped the quasi-mode structure through the radiation of harmonic oscillator. sheared VRWs, in a manner similar to that described If we now view the vertical shear as a time-invariant by RM01 for general perturbations in the QG context. forcing [cf. Eq. (12)], Eq. (20) can be generalized to Thus, at large Rossby number and for simple tilts, while that of a forced damped harmonic oscillator. For zonal the quasi-mode decay accounts for the latter portion of vertical shear ¯ow the associated PV has a horizontal the overall tilt decay on the time scales presented, the gradient in the y direction, and the forcing on the rhs bulk of the alignment takes place rapidly at early times is of the form 2Fq,s(r, z) cos␭. The solution to the forced through the axisymmetrization of PV via the sheared equation is

VRW mechanism. i␭ 2Fq,s e qЈ(r, ␭, z, t) ϭ Re A(r, z)ee␥ti(␭Ϫ␻pt) ϩ , i␻ Ϫ ␥ 2 []p 4. A heuristic model for a distributed vortex in vertical shear (21) where A(r, z) is a complex amplitude. Assuming the From the linear VRW perspective there are three ways initial vortex is vertically aligned, that is, qЈ(t ϭ 0) ϭ in which the vortex in shear remains vertically aligned. 0, and utilizing the fact that | ␥/␻p | K 1, it follows In the absence of a quasi mode, the tilt perturbation from (21) that generated by differential advection of the vortex by the shear ¯ow will disperse on the mean vortex as sheared F (r, z) qЈ(r, ␭, z, t) ഠ Ϫ q,s [e␥t sin(␭ Ϫ ␻ t) Ϫ sin␭]. VRWs. If the axisymmetrization of the perturbation PV ␻ p (i.e., the tilt) can counter the production by differential p (22) advection, then the vortex will remain upright. When a vortex supports a quasi mode, we found in the previous Therefore, for small but ®nite | ␥/␻p | the long-time so- section that the tilt evolution in the unforced case is lution will always be one of ®xed tilt, oriented perpen- actually a superposition of the sheared VRWs and the dicular to the direction of shear, or downshear-left. The quasi mode. In this case the resulting evolution should downshear-left con®guration is optimal because the ver- re¯ect both parts of the solution, but may be dominated tically penetrating ¯ow associated with the tilted PV of by the latter if it is more robust. In the free alignment the vortex has a vertical shear opposite in sign to the example of Fig. 5 the quasi mode was observed to decay environmental vertical shear ¯ow (SJ95). For stable vor- at a much slower rate than the transient sheared VRWs. tex pro®les (i.e., ␥ Յ 0), the counteracting vertical It stands to reason, then, that the forced solution will shears across the vortex core make the equilibrium so- tend to ``lock on'' to the less evanescent quasi-modal lution a possibility. Figure 6a illustrates schematically part. If the quasi mode dominates the evolution, then the vortex tilt evolution in this case. It should be noted, the vortex resists tilting by vertical shear in two ways, however, that the basis for assuming solutions of the through precession upshear (similar to the mechanism form given by Eq. (19) becomes more tenuous as | ␥/ described by SJ95) and resonant damping. Regardless ␻p | approaches unity. In this regime SMR demonstrated of whether or not a quasi mode exists, the mechanism that the quasi mode rapidly disperses on the mean vortex for vortex tilt reduction is VRW damping. as sheared VRWs, and thus this simple harmonic os- In that part of the free-alignment phase space where cillator analogy no longer holds. decaying quasi modes exist, Schecter and Montgomery Recall from section 3 that in the TC regime (L/lr,G ϳ (2003) found the vortex tilt PV takes the form of a 0.1) the tilt of a Gaussian vortex supporting a quasi damped discrete VRW (for n ϭ 1): mode decays at a very slow rate on the time scale of the vortex precession. In this case ␥ ഠ 0, and we arrive ␥ti(␭Ϫ␻ t) qЈ(r, ␭, z, t) ϭ Re[C(r, z)ee p ], ␥ Ͻ 0, (19) at the solution for a forced undamped harmonic oscil- where C(r, z) provides the radial and vertical structure lator: of the tilt, ␥ is the exponential damping rate, and ␻p is Fq,s(r, z) the vortex precession frequency. In the derivation of Eq. qЈ(r, ␭, z, t) ϭϪ [sin(␭ Ϫ ␻pt) Ϫ sin␭]. (23) ␻p (19) it is assumed that | ␥/␻p | K 1. The temporal part of Eq. (19) is the solution to a differential equation of For all values of ␻p this solution represents the perpetual the form tilting, precession, and realignment of a vortex in zonal 1JANUARY 2004 REASOR ET AL. 13

FIG. 6. Schematic illustration of the vortex tilt evolution [based on Eq. (22)] for the case where (a) the VRW

damping rate ␥ is a nonnegligible fraction of the precession frequency ␻p and (b) the VRW damping rate is zero. The initially aligned vortex is tilted by westerly vertical shear. The vortex center at upper levels is denoted by the stars and at lower levels by the hurricane symbols. The thin arrows indicate the direction of motion of the upper- and lower-level centers. The vortex in (a) achieves a steady-state tilt to the left of the vertical shear vector. In (b) the vortex tilts downshear, precesses cyclonically upshear, realigns, and then repeats this evolution. shear. Figure 6b depicts the vortex tilt evolution in the tex vertically aligned when a vertical shear ¯ow is im- limit of in®nitesimal damping. posed, we continue our investigation using the simple This linear harmonic oscillator model of the vortex linearized PE model forced by a time-invariant zonal in shear breaks down once the vertical tilt becomes suf- vertical shear ¯ow. As described in section 2, the shear ®ciently large. Smith et al. (2000) showed that, as the ¯ow is chosen such that it projects only onto the ®rst vertical shear is increased, an idealized vortex can be internal vertical mode, and thus forces a vertical tilt made to transition from a regime where the vortex pre- similar to that given by Eqs. (15)±(16). Any complete cesses to one where the upper- and lower-level PV study of TC-like vortices in vertical shear must ulti- anomalies of the vortex separate inde®nitely [the Smith mately permit the shearing apart of the vortex as a pos- et al. model does not, however, permit initially aligned sible solution, which the linear model obviously does vortices to achieve steady-state solutions of the form not allow. But in the parameter regime for which the given by Eq. (22)]. The irreversible separation of upper- linear approximation is reasonably accurate (i.e., over- and lower-level PV anomalies is fundamentally nonlin- lapping upper- and lower-level PV cores), it serves as ear, and occurs when the vortex is unable to precess a useful ®rst approximation for understanding the basic quickly enough to overcome the differential advection dynamics. For the idealized vortex simulations pre- by the shear ¯ow. In the upcoming discussion we char- sented below all linear results are compared against sim- acterize the ``breaking point'' of the vortex in shear by ulations using the fully nonlinear dry dynamical core the time it takes the horizontal distance between upper- of the CSU±RAMS model. and lower-level PV centers to increase by twice the RMW, 2L, under simple advection by the shear ¯ow. a. Vortex resiliency in shear For the shear ¯ow de®ned by Eq. (9) the differential Ϫ1 Consider a weak vortex (V ϭ 10msϪ1, L ϭ 100 advection rate ␶ s ϳ U/L. max km) with Ro ϭ 3.2 and L/lr,G ϭ 0.08 in easterly vertical shear of4msϪ1 (U ϭ 2msϪ1) over a 10-km depth. 5. Vortex alignment in vertical shear: Numerical According to Fig. 3, the vortex exhibits little tendency simulations for alignment as it is dominated by a weakly decaying

To further understand how the VRW damping mech- quasi mode. In this case the precession frequency ␻p ϭ anism described in the previous sections keeps the vor- 5.26 ϫ 10Ϫ6 sϪ1 is an order of magnitude less than the 14 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

FIG. 7. Evolution of an initially upright Gaussian vortex in easterly vertical shear of 4 m Ϫ1 Ϫ1 s (10 km) . Shown are the locations of the PV maxima every ␶e/4 at z ϭ 0 (hurricane

symbols) and z ϭ 10 km (stars). The vortex is of tropical depression strength (Vmax ϭ 10 m Ϫ1 s ) with Ro ϭ 3.2 and L/lr,G ϭ 0.08, placing it well within the quasi-mode regime of Fig. 3. The initial shearing apart of the vortex is well captured by the linear PE model (solid). A long-time shearing apart of the vortex is predicted by the nonlinear RAMS model (dotted).

Ϫ1 Ϫ5 Ϫ1 differential advection rate ␶ s ϭ 2 ϫ 10 s . The evolution of the upper- and lower-level PV maxima shown in Fig. 7 re¯ects the relatively slow precession of the quasi mode. Little meridional displacement compared to zonal displacement is observed. In the RAMS simulation the zonal displacement, while in agreement with the basic trend shown in the linear simulation at early times, is much more dramatic, with the vortex

completely shearing apart after t ϭ 1␶e. The linear solution is still useful because it suggests that, when ␥ Ϫ1 and ␻p are much less than␶ s , the vortex will tend to shear apart. Keeping all else the same, but increasing Ro to 12.7 Ϫ1 (Vmax ϭ 40ms , L ϭ 100 km), a very different vortex evolution is observed. Figure 8 shows time series of the azimuthal wavenumber-1 PV amplitude (normalized by the maximum mean PV of the vortex) and phase associated with the tilt asymmetry. As in the small Rossby number case, the vortex ®rst tilts in the direction of the vertical shear. The PV perturbation in this case, however, never reaches a signi®cant fraction of the mean PV. Thus, the vortex remains nearly aligned at all times, and the nonlinear RAMS solution is well approximated by the linear solution. The vortex precesses upshear, realigns, and then repeats the process over again. Such an evolution was anticipated in the previous section Ϫ1 when ␥ is negligible and ␻p Ͼ ␶ s . Here ␻p ϭ 8 ϫ Ϫ5 Ϫ1 Ϫ1 Ϫ5 Ϫ1 10 s , which is larger than ␶ s ϭ 2 ϫ 10 s . The FIG. 8. Amplitude (normalized by the maximum mean PV) and tilt decay at long times is approximately zero, so the phase of the azimuthal wavenumber-1 PV asymmetry for the hurricane-strength (40 m sϪ1) Gaussian vortex. All parameters are as in conditions for the forced undamped harmonic oscillator Fig. 7, except Ro ϭ 12.7. The vortex remains vertically coherent by are satis®ed. The forced undamped solution (23) is plot- precessing upshear (180Њ) and realigning. The linear evolution (solid) ted in Fig. 8. It agrees well with the linear PE solution, agrees well with the nonlinear RAMS simulation at z ϭ 0 (dashed) as expected. and z ϭ 10 km (dotted), and also with the forced undamped harmonic oscillator solution based on Eq. (23) (dashed±dotted). For simplicity Suppose we now increase the vertical shear in this we have normalized the harmonic oscillator solution to match the large Rossby number case. How strong must the shear maximum amplitude of the linear PE solution. get before the vortex shears apart, and what determines 1JANUARY 2004 REASOR ET AL. 15

that threshold? Is it simply a matter of the quasi mode not being able to precess quickly enough? Using the RAMS model, we illustrate in Fig. 9 the changes in vortex behavior with increasing vertical shear. At 8 m sϪ1 over a 10-km depth, the upper- and lower-level vortex PV remains vertically aligned in the RAMS simulation, precessing as in the weaker shear case but at a slightly slower cyclonic speed. The linear precession frequency (ϭ8 ϫ 10Ϫ5 sϪ1) is still found to be twice the differential advection rate (ϭ4 ϫ 10Ϫ5 sϪ1), so the continued resiliency of the vortex is not surprising based on our simple arguments. The actual tilt of the vortex PV is shown in Fig. 10. When the vortex reaches its maximum tilt at 90Њ, the horizontal separation between upper- and lower-level PV maxima reaches approximately 60 km (Fig. 10a). As also noted by SJ95, the tilt of the inner core of the vortex is noticeably less than that of the larger-scale vortex. Since the vortex evolution is largely balanced, one need not invoke the divergent part of the circulation to explain this structureÐthe transverse circulation exists to maintain balance (RM01). Consistent with our VRW theory for vortex alignment, the radial vortex structure is, instead, a consequence of the emergence of the quasi mode. As previously discussed [see section 3c(3)], the quasi mode at ®nite Rossby number does not take the form of a simple, spatially uniform tilt perturbation (as it does in the QG theory). Thus, the superposition of the mean vortex and FIG. 9. As in Fig. 8 but with stronger easterly vertical shear of 8msϪ1 (10 km)Ϫ1. the emerging quasi mode will yield a vortex tilt that is spatially nonuniform, as shown in Fig. 10. As the vortex continues to precess upshear, both the vortex core and large-scale vortex realign (Fig. 10b).

FIG. 10. Vertical cross section of Ertel PV in the direction of tilt (90Њ) at (a) t ϭ 3␶e (maximum

tilt) and (b) t ϭ 6␶e (realignment) for the nonlinear RAMS simulation of Fig. 9. Values are in PVU, where 1 PVU ϭ 10Ϫ6 m2 sϪ1 KkgϪ1. Notice the tilt of the vortex core is smaller than that of the larger-scale vortex. 16 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

When the vertical shear equals 12 m sϪ1 (10 km)Ϫ1 the vortex appears to begin to shear apart at early times, but subsequently realigns and stays vertically coherent (not shown). A new phenomenon arises at this larger value of shear not seen at 8 m sϪ1 (10 km)Ϫ1.Asthe vortex tilt becomes large and the local vertical shear of the vortex winds also increases, the gradient Richardson -z)2], drops beץ/␷ץ) z) 2 ϩץ/uץ)]/number, de®ned as N 2 low unity in localized regions of the vortex core. A local instability occurs within these regions, veri®ed with horizontal grid spacing of ⌬x ϭ⌬y ϭ 10 km and ⌬x ϭ ⌬y ϭ 2 km. In both cases the vertical grid spacing ⌬z ϭ 1 km. The PV in these localized regions becomes large, implicating an upgradient transfer due to subgrid- scale processes [cf. Thorpe and Rotunno (1989) section 2]. We speculate that the mixing that accompanies this instability is not well resolved at the present RAMS vertical resolution. The instability ultimately ceases, though, and the vortex relaxes to a smooth, vertically coherent structure. Further investigation of this behavior for strong vortices in strong shear will be reported upon in a forthcoming publication. b. Comparison with previous work The results presented here using the vortex given by Eq. (14) differ from those of SJ95. The large Rossby number simulation depicted in Fig. 8 uses vortex and environmental parameters very similar to those used by SJ95 in her benchmark simulation. When embedded in 4msϪ1 (10 km)Ϫ1 vertical shear, the tilt of her vortex increased steadily throughout the duration of the 96-h simulation. Our vortex remains vertically coherent with only a small departure from an aligned state at any time. FIG. 11. The azimuthal-mean vortex used in the Hurricane Olivia 1 The primary difference between the two simulations is (solid) and SJ95 (dotted) simulations. Also shown is the 40 m s Ϫ Gaussian vortex used in Fig. 8 (dashed). The tangential wind and the radial structure of the initial vortex. The vortex pro- relative vorticity are in units of m sϪ1 and 10Ϫ3 sϪ1, respectively. The ®le used by SJ95 is shown in Fig. 11. Although her relative vorticity inset shows the radial structure of the SJ95 vortex vorticity pro®le closely resembles our monotonically outside the RMW. The Rossby number, Ro ϭ ␷ (r)/ fr, is based on decreasing Gaussian within the RMW, outside the ap- the azimuthal mean tangential velocity of the untilted vortex. proximately 170-km radius the relative vorticity becomes negative and then asymptotes to zero at large radius. Similar barotropic pro®les used by Gent and SJ95 out to approximately t ϭ 3␶e, as shown in Fig. McWilliams (1986) were found to support an internal 12b (compare with the SJ95 Fig. 2). instability in the QG system, which could be interpreted It is tempting to speculate on the cause of the ex- in our problem as a ``tilt instability.'' ponential tilt growth in the SJ95 free alignment simu- To test whether the susceptibility of the SJ95 bench- lation based on the recent extension of resonant damping mark vortex to vertical shear could simply be a con- theory to ®nite Rossby number by Schecter and Mont- sequence of this tilt instability generalized to ®nite Ross- gomery (2003). According to the resonant damping the- by number, we perform a free alignment simulation us- ory [i.e., Eq. (19)], the growth/decay rate ␥ of the tilt ing the linear PE model as in section 3, but with the perturbation is proportional to the mean PV gradient at SJ95 pro®le. The results are shown in Fig. 12a. A tran- the critical radius in the ¯ow where the precession fre- sient decay of the vortex tilt is quickly followed by quency equals the vortex rotation. For the Gaussian vor- exponential growth through the duration of the simu- tex ␥ is always negative, and damping occurs. But for lation with a growth rate of 2.2 ϫ 10Ϫ5 sϪ1. Thus, even the SJ95 vortex it is possible for ␥ to be positive, and without vertical shear the vortex is prone to shearing hence for resonant growth to occur. Upon calculating apart. When the 4 m sϪ1 (10 km)Ϫ1 vertical shear is the precession frequency of the growing tilt perturbation Ϫ5 Ϫ1 added in the linear PE model, the results are quite similar in the SJ95 simulation (␻p ϭ 4.7 ϫ 10 s ), we ®nd to the nonlinear PE benchmark simulation results of that the location in the ¯ow where resonance would 1JANUARY 2004 REASOR ET AL. 17

FIG. 12. Linear PE evolution of the benchmark vortex described in SJ95 subjected to easterly vertical shear of4msϪ1 (10 km)Ϫ1. (a) Loglinear plot of vortex tilt from the unforced (dashed) and forced (solid) simulations (normalized by the maximum unforced value). The initial tilt amplitude in the unforced simulation is ␣ ϭ 0.3. The vortex is unstable to small tilt perturbations. (b) Total Boussinesq vortex PV at z ϭ 0 (solid) and z ϭ 10 km (dashed) (contour interval 0.4 ϫ 10Ϫ7 sϪ3). occur falls within the region of positive PV gradient at they are allowed to remain as robust features outside approximately 300-km radius (see Fig. 11; recall dq/dr the vortex core. By broadening the vortex pro®le (and ϭ N 2d␨/dr). Further examination of the possible link thus moving the region of negative relative vorticity between the tilt instability of Gent and McWilliams farther from the vortex core), Jones (2000a) showed that (1986) and resonant growth is reserved for a future pub- the vortex remains vertically coherent at long times, lication. behaving as our sheared hurricane-strength Gaussian In Jones (2000a) the benchmark vortex pro®le of SJ95 vortex. We have veri®ed using our linear model that the was used in a simulation similar to that described above, critical radius for the vortex in this case (see her Fig. but at higher latitude. The vertical tilt initially increased 13, case NP2) falls in a region of small but negative as the vortex precessed upshear, but then decreased as mean vorticity gradient, and thus does not support a tilt the vortex achieved a downshear orientation. Jones instability. (2000a) showed that this behavior results from advection of the vortex core by large-scale asymmetries gen- c. Hurricane Olivia (1994) erated by the vertically penetrating ¯ow of the tilted vortex. These large-scale asymmetries only have sig- As our ®nal example, we apply the VRW damping ni®cant impact as the vortex tilt becomes large and when ideas developed in the foregoing to the case of Hurricane

FIG. 13. Linear PE evolution of the Olivia vortex (Ro ϳ 68 and L/lr,G ϳ 0.02) subjected to westerly vertical shear of 10 m sϪ1 (10 km)Ϫ1. (a) Vortex tilt from the unforced (dashed) and forced (solid) simulations (normalized by the maximum unforced value). The initial tilt amplitude in the unforced simulation is ␣ ϭ 0.3. (b) Azimuthal wavenumber-1 component of Boussinesq PV at z ϭ 0 (contour interval 0.5 ϫ 10Ϫ7 s Ϫ3). Stars denote the phase evolution predicted by the forced damped harmonic oscillator solution (see text for details). The long- term solution is a steady-state tilt (south to north with height) to the left of the westerly shear vector. 18 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

Olivia (1994). The vortex pro®le shown in Fig. 11 is phase evolution is approximately reproduced, as indi- taken from the observational study of Olivia by Reasor cated in Fig. 13b. et al. (2000). Olivia was observed to weaken from ``top down'' as north-northwesterly vertical shear increased 6. Summary and discussion from 2±3 m sϪ1 to 15 m sϪ1 over the lowest 10-km depth of the storm in a 4-h period. The vortex tilt in We have presented a new theory for the resiliency of the lower to middle troposphere was approximately 2± initially barotropic and stable TC-like vortices in ver- 3 km with a west-to-east orientation with height. Oli- tical shear ¯ow, demonstrating the fundamental role of via's wind structure before encountering the large ver- VRWs in the maintenance of vortex vertical alignment tical shear consisted of a prominent barotropic com- and the remarkable ability of such vortices to withstand ponent. For simplicity, we assume the mean vortex is strong vertical shear (Ͼ10msϪ1 over the depth of the initially barotropic. We have also arti®cially ®lled the troposphere) without the direct aid of cumulus convec- eye with cyclonic vorticity in order to eliminate the tion. Previous studies have invoked the concept of pen- emergence of barotropically unstable modes, discussed etration depth to account for the ability of dry TC-like by Reasor et al. (2000). Using H ϭ 10 km, f ϭ 4.5 ϫ vortices to ®ght differential advection by vertical shear, Ϫ5 Ϫ1 Ϫ2 Ϫ1 and thus remain vertically coherent. While this view is 10 s , N ϭ 1.22 ϫ 10 s , L ϭ 17 km, and Vmax Ϫ1 both physical and conceptually appealing, it is incom- ϭ 52 m s yields Ro ϭ 68 [note that here Ro ϭ Vmax/ fL, whereas Schecter and Montgomery (2003) de®ne plete. It cannot explain, for example, the subtle dependence of the TC evolution on radial pro®le as demon- Ro ϭ ␨(0)/ f] and L/lr,G ϭ 0.02. Although Fig. 3 is strictly valid for Gaussian-like vorticity pro®les, it sug- strated here, the presence of quasi-stationary tilted states gests that, if the results were extrapolated to very large for the TC in shear (e.g., downshear-left), and the prom- Rossby number, the Olivia vortex might support a quasi inence of radially and azimuthally propagating VRWs mode. In Fig. 13a we show the free alignment of the that arise when a strong TC is forced by vertical shear. Olivia vortex. It is dif®cult to discern the presence of The theory presented here sheds light on all of these a quasi mode given the high amplitude oscillations in issues, exposing clearly the quasi-elastic nature of a vortex tilt. Schecter and Montgomery (2003) have in- stable TC and the VRWs that are intrinsic to such a vortex. deed veri®ed the presence of a rapidly decaying quasi Vertical shear acts as a VRW generator when it in- mode for the Olivia case (see their Fig. 11). Due to its teracts with a TC. To illustrate this we began by con- rapid decay with time, however, it is unclear whether it sidering the free alignment of ®nite Rossby number vor- will dominate the forced solution in the way the quasi tices. Given a simple vertical tilt, a vortex with mono- mode apparently dominated the evolution of the large tonically decreasing mean PV will align through the Rossby number Gaussian vortex in vertical shear. projection of the tilt asymmetry onto stable VRWs. Vor- When westerly vertical shear of 10 m sϪ1 (10 km)Ϫ1 tices for which L/Lr,G K 1 are likely to support a near- is imposed, Fig. 13a shows that the vortex tilt remains discrete VRW, or quasi mode, especially at small Rossby small for the duration of the simulation, never quite number. The quasi-mode propagation causes the vortex reaching the initial small tilt of the free alignment ex- to precess, and its decay leads to slow alignment. Schec- Ϫ1 Ϫ4 Ϫ1 periment. In this case ␶ s ϭ 2.94 ϫ 10 s , less than ter and Montgomery (2003) have developed a theoret- Ϫ4 Ϫ1 the precession frequency ␻p ϭ 3.9 ϫ 10 s deter- ical formalism for studying the quasi mode at ®nite mined by Schecter and Montgomery (2003). Thus, we Rossby number by which the linear damping rate and would expect the linear results to be at least qualitatively precession frequency can be determined. We have dem- accurate. According to Fig. 13b, the vortex tilts from onstrated, however, that not all vortices support quasi west to east with height at early times, consistent with modes within the parameter regime typical of TCs. Spe- the direction of the vertical shear. The vortex precesses ci®cally, a vortex for which the horizontal scale is a cyclonically, as in previous simulations but only rotates nonnegligible fraction of the global internal deformation through 90Њ, reaching a near-steady-state tilt con®gu- radius and Ro k 1 will instead align through the dis- ration downshear-left of the vertical shear.3 This type of persion of the vortex tilt asymmetry on the mean vortex evolution was predicted in section 4 for the case of as sheared VRWs. The inviscid damping of the sheared nonnegligible quasi-mode decay rate. Inserting the val- VRWs then leads to rapid reduction of the vortex tilt. Ϫ5 Ϫ1 Ϫ4 Ϫ1 ues of ␥ (Ϫ4.86 ϫ 10 s ) and ␻p (3.9 ϫ 10 s ) When vertical shear is applied to an initially aligned determined by Schecter and Montgomery (2003) into vortex, the TC not supporting a quasi mode will counter the forced damped harmonic oscillator Eq. (22), the PV the production of vertical tilt by differential advection through sheared VRW dispersion. On the other hand, the evolution of a TC-like vortex supporting a quasi 3 The linear PE results have been veri®ed using the nonlinear mode was shown to be dominated by the quasi mode RAMS model. Although RAMS is non-Boussinesq, and thus the PV is not symmetric about middle levels as in the linear Boussinesq PE when it is forced. We know from Schecter and Mont- model, the vortex still still achieves a steady-state tilt oriented down- gomery (2003) that the free-alignment solution is simply shear-left. that of a damped harmonic oscillator in the limit of small 1JANUARY 2004 REASOR ET AL. 19 damping rate ␥ compared to the precession frequency tilting instability, causing growth of the tilt with time,

␻p. If the shear forcing is time invariant, the resulting even as the vortex precesses. This early evolution is, in forced solution is straightforward. fact, consistent with our forced harmonic oscillator The precise behavior of the vortex supporting a quasi model [Eq. (22)], but with ␥ Ͼ 0. Although quantitative mode in shear depends on the radial location where ␻p mesoscale observations of hurricane tilt are limited equals the angular rotation rate of the mean ¯ow, what (Marks et al. 1992; Reasor et al. 2000), and to date few Schecter and Montgomery (2003) de®ne as the ``critical studies have focused on hurricane tilt evolution in full- radius.'' The damping rate is proportional to the radial physics numerical simulations (Wang and Holland 1996; gradient of mean PV at this radius. A Gaussian vortex Frank and Ritchie 1999, 2001), the consensus thus far is characterized by weak PV gradients outside the vortex appears to be that hurricanes do not precess continually core, so in the TC regime ␥ is typically negligible.4 For when forced by vertical shear [although they may wob- meteorologically relevant timescales (t Յ 1±2 days), the ble due to other mechanisms not discussed here (e.g., Gaussian vortex in shear behaves as a forced undamped Nolan et al. 2001)]. When a hurricane is forced by ver- harmonic oscillator characterized by the perpetual tilt- tical shear, the observations suggest that the vortex tilt ing, cyclonic precession upshear, and realignment of the is generally oriented downshear to downshear-left vortex. Studies of realistic hurricanes in vertical shear (Marks et al. 1992; Reasor et al. 2000). This orientation typically do not observe such an evolution. The reason is consistent with our dry adiabatic results when the for this is that a hurricane pro®le, unlike the Gaussian, decay rate of the quasi mode is nonnegligible. It then typically exhibits large radial PV gradients at 2±3 times stands to reason that the downshear-left location of con- the RMW where the critical radius is likely to fall, lead- vective asymmetry observed in hurricanes forced by ing to more rapid decay of the quasi mode. A heuristic vertical shear may be in the ®rst approximation a con- analysis showed that the solution in this case is one of sequence of the dry adiabatic VRW dynamics. steady-state tilt oriented left of the vertical shear vector. Having clari®ed how the tilt of vertically sheared TC- We con®rmed this behavior via a linear PE numerical like vortices is reduced, the problem of vortex resiliency simulation using the observed wind pro®le of Hurricane was then addressed through comparison of linear PE Olivia (1994). It should be noted that this evolution is and dry nonlinear RAMS simulations of TC-like vor- also consistent with that of Wang and Holland (1996, tices in vertical shear. In the absence of a strong align- their Fig. 4) who used numerically simulated hurricane ment mechanism a TC in vertical shear will be differ- pro®les. Although they invoked the out¯ow anticyclone entially advected by the mean ¯ow, causing the vortex to explain the steady-state tilt con®guration, here we to tilt irreversibly. The linear PE model captures such show that the cyclonic portion of the vortex alone can an evolution until the upper- and lower-level PV cores explain the downshear-left tilt orientation. cease to overlap horizontally. We de®ned a characteristic

We believe that our demonstration of TC-like vortices time scale for this threshold, ␶s ϳ 2L/⌬U, where 2L is achieving approximate steady-state tilts to the left of the twice the RMW and ⌬U is the difference in shear ¯ow shear vector in the dry adiabatic model is signi®cant. between upper and lower levels of the vortex. For vor- While the downshear-left tilt orientation is the one that tices supporting a quasi mode the RAMS simulation minimizes the net vertical shear (i.e., the superposition con®rmed the shearing apart of the vortex when ␻p K Ϫ1 Ϫ1 of the vertically penetrating ¯ow of the tilted vortex and ␶␶ss. When ␻p Ͼ , the vortex is able to remain ver- the environmental vertical shear ¯ow), and thus would tically aligned through precession upshear. The linear appear to be an optimal con®guration, the possibility of solution remains valid in this regime. For a 40 m sϪ1 the downshear-left solution actually depends on the de- Gaussian vortex with L ϭ 100 km (and lr,G ϭ 1230 km) tails of the vortex pro®le. Previous studies using ide- we found that the vortex remains vertically aligned at alized pro®les have given the impression that the pre- shears up to 12 m sϪ1 over 10-km depth. For the Hur- cessing solution is a typical one in the small tilt limit. ricane Olivia (1994) pro®le the vortex remained verti- The precession shown by Smith et al. (2000), for ex- cally coherent in 10 m sϪ1 (10 km)Ϫ1 vertical shear. ample, is a consequence of their point vortex model not Based on these results, we conclude that the dry adia- permitting decaying quasi-mode solutions. For the SJ95 batic dynamics is capable of maintaining the vertical vortex we have shown that the vortex supports a linear alignment of strong, approximately barotropic stable vortices in shear greater than 10 m sϪ1 over the depth of the vortex. 4 For suf®ciently large Ro, vortices with weak vorticity gradients To determine precisely what sets the critical threshold outside their core are in principle susceptible to an exponential Ross- by±inertia buoyancy (RIB) instability that involves a resonance be- between alignment of a TC in shear and the irreversible tween the VRW in the core and an inertia buoyancy wave in the shearing apart of the vortex, we require a clearer un- 2 ⍀ / derstanding of the role of nonlinear processes in vortexץt ϩץ/ץ) environment. Formally, this can occur when DI ϭ -␭)2␩␰ Ͼ 1 (Schecter and Montgomery 2003). For all of the nu- alignment at ®nite Rossby number (Ro Ͼ 1). The comץ merical experiments shown here, we do not observe the RIB insta- Ϫ1 bility. The vorticity gradient at the critical radius is evidently suf®- parison of ␻p and␶ s is only a crude measure of that ciently negative to suppress the RIB instability [see Schecter and threshold. The impact of baroclinic vortex structure on Montgomery (2003) for details]. the resiliency of TCs must also be examined since TC 20 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 61

winds are observed to decay up to the tropopause. A ported in part by the National Research Council while recent idealized study of baroclinic vortices in vertical a postdoctoral research associate at the Hurricane Re- shear by Jones (2000b) illustrates how the weaker vortex search Division of AOML/NOAA. The authors would circulation at upper levels is especially prone to being like to thank Drs Sarah Jones, Mark DeMaria, Yuqing stripped away from the rest of the cyclonic circulation Wang, and Ray Zehr for their helpful discussions re- below it. How the VRW damping mechanism presented garding this work. They also wish to thank two anon- here might shed light on this behavior is currently under ymous reviewers for their constructive comments on an investigation. earlier version of this paper. In light of the present ®ndings, the impact of diabatic processes on TC resiliency needs closer examination. APPENDIX A Diabatic processes aid the alignment of TCs indirectly through reduction of the static stability in regions of RAMS Core moist ascent and intensi®cation of vortex PV via the low- to midlevel convergence of azimuthal-mean vor- For the dry dynamical core of RAMS, the vector ticity by the axisymmetric component of the diabatically momentum, perturbation Exner function, and pertur- driven secondary circulation. The former will tend to bation potential temperature equations are, respectively, increase L/Lr,G, and the latter will increase the charac- dv (teristic Rossby number and the radial gradient of azi- ϭ ␪0١␲ЈϩBk Ϫ fk ϫ v ϩ Fm, (A1 muthal-mean PV. A decrease in the Rossby radius of dt ␲Ј c2ץ deformation reduces the intrinsic VRW propagation (␳␪v), (A2)´ ١ ϭϪ s t ␳␪2 00ץ (speed and leads to a greater absolute (ground based azimuthal propagation speed (Reasor et al. 2000). An 00 increase in Rossby number also will increase the ab- d␪Ј ϭ F , (A3) solute azimuthal VRW propagation speed, or in the pres- dt ␪ ent vortex alignment context, the precession frequency of the vortex. We have demonstrated that an increased where v is the vector wind, ␲Ј is the perturbation Exner precession frequency makes the vortex more resilient to function de®ned as the difference between the Exner ␬ differential advection by the vertical shear ¯ow. The function ␲ ϭ cp(p/p 0) and the reference state Exner increased precession frequency also shifts the critical function ␲ 0, B ϭ g␪Ј/␪ 0 is the buoyancy, and ␪Ј is the radius radially inward into a region of larger negative perturbation potential temperature. Here ␳ 0, ␪ 0, and p 0 mean PV gradient (which itself may be enhanced by are the reference state density, potential temperature,

diabatic processes). The increased VRW damping rate and pressure, respectively, and cs is the adiabatic speed makes the vortex even more resilient to vertical shear. of sound. Subgrid-scale processes are represented by Fm

Diabatic processes, however, may not always have a in the momentum equation and by F␪ in the potential positive impact on vortex alignment. The numerical temperature equation. simulations of Frank and Ritchie (2001) suggest that For the TC-in-shear simulations the RAMS model diabatic processes can have a deleterious impact on ma- was run nonhydrostatically and compressibly (Pielke et ture hurricanes in strong vertical shear through the con- al. 1992). Momentum was advanced using a leapfrog vective asymmetry generated by the vortex±shear in- scheme, while scalars were advanced using a forward teraction. Subsequent to the development of a pro- scheme; in addition, both methods used second-order nounced convective asymmetry, they found that a top- advection. RAMS uses the Arakawa fully staggered C down weakening of the hurricane generally ensues. grid (Arakawa and Lamb 1981). Perturbation Exner Understanding how diabatic heating modi®es the evo- function tendencies, used to update the momentum var- lution of the quasi mode and the sheared VRWs and iables, were computed using a time split scheme similar why the convective asymmetry of the tilted hurricane to Klemp and Wilhelmson (1978). Lateral boundaries can in some cases hinder alignment via the VRW damp- used the Klemp±Wilhelmson condition; that is, the nor- ing mechanism is an important topic worthy of future mal velocity component is speci®ed at the lateral bound- study. ary and is effectively advected from the interior.

Note added in proof: Our simulations in section 5e Subgrid-scale processes Fm were parameterized using use the tangential wind pro®le given in Eq. (1) of SJ95, Fickian diffusion throughout the bulk of the domain which has since been discovered to contain a typograph- with a diffusion coef®cient of 1000 m2 sϪ1. For this ical error. The impact of this error on the pro®le is small, value of the diffusion coef®cient we observed good and should not affect the basic results of section 5c. agreement between linear inviscid and RAMS simulations. In the vortex core the Smagorinsky deformation- Acknowledgments. This work was supported in part based eddy viscosity (Smagorinsky 1963) was activated by Of®ce of Naval Research Grant N00014-02-1-0474, if values exceeded the minimum Fickian value. In all National Science Foundation Grant ATM-0101781, and simulations shown the computed value of vertical dif- Colorado State University. The ®rst author was sup- fusion was zero. 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