progress in atmospheric vortex structures deduced from...

3 February 2009 9:2 B-582 ch16

Progress in Atmospheric Vortex StructuresDeduced from Single Doppler Radar Observations

Wen-Chau Lee

Earth Observing Laboratory, National Center for Atmospheric Research∗,Boulder, Colorado, USA

[email protected]

Ben Jong-Dao Jou

Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan

Doppler radars have played a critical role in observing atmospheric vortices including tor-nados, mesocyclones, and tropical cyclones. The detection of the dipole signature of a meso-cyclone by pulsed Doppler weather radars in the 1960s led to an era of intense researchon atmospheric vortices. Our understanding of the internal structures of atmospheric vor-tices was primarily derived from a limited number of airborne and ground-based dual-Doppler datasets. The advancement of single Doppler wind retrieval (SDWR) algorithmssince 1990 [e.g. the velocity track display (VTD) technique] has provided an alternateavenue for deducing realistic and physically plausible two- and three-dimensional structuresof atmospheric vortices from the wealth of data collected by operational and mobile Dopplerradars.

This article reviews the advancement in single Doppler radar observations of atmospheric vor-tices in the following areas: (1) single Doppler radar signature of atmospheric vortices, (2) SDWRalgorithms, in particular the VTD family of algorithms, (3) objective vortex center-finding algo-rithms, and (4) vortex structures and dynamics derived from the VTD algorithms. A newparadigm that improves the VTD algorithm, displaying and representing the atmospheric vor-tices in VdD/RT space, is presented. The VTD algorithm cannot retrieve the full componentsof the divergent wind which may be improved by either implementing physical constraints onthe VTD closure assumptions or combining high temporal resolution data with a mesoscale vor-ticity method. For all practical perspectives, SDWR algorithms remain the primary tool for ana-lyzing atmospheric vortices in both operational forecasts and research purposes in the foreseeablefuture.

∗The National Center for Atmospheric Research is sponsored by the US National Science Foundation.

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Progress in Atmospheric Vortex Structures Deduced from Single Doppler Radar Observations 327

1. Introduction

Atmospheric vortices, such as tornados, meso-cyclones, and tropical cyclones (TCs), are highlycorrelated with severe weather and disastersthreatening human lives and property. Famousexamples are Typhoon Herb (1996), HurricaneAndrew (1992), Hurricane Katrina (2005), andthe supertornado outbreak on 3–4 April 1974in the United States, where hundreds of livesand billions of dollar’s worth of property werelost because of damaging winds, mudslides, andfloods. The spatial and temporal scales of thesevortices span five orders of magnitude (from tensof meters to hundreds of kilometers, and fromseveral minutes to ten days).

Sampling and deducing the two-dimensionaland three-dimensional internal structures ofthese wide ranges of atmospheric vortices havebeen ongoing challenges for operational fore-casts, scientific research, and numerical modelinitialization and validation (e.g. Houze et al.,2006; Wang and Wu, 2004).

Atmospheric vortex structures have beenobserved by and inferred from in situ measu-rements, remote sensing, and photogrammetry(e.g. Davies-Jones et al., 2001; Marks, 2003;Gray, 2003). Doppler radar remains the onlyinstrument that can probe 3D internal struc-tures of cloud and precipitation systems at aspatial resolution of 0.025–1km and a temporalresolution of 1–30 minutes. Since Doppler radarssample only one component of the 3D motionalong a radar beam, estimating 3D wind vectorsrequires two or more Doppler radars viewing thearea of interest simultaneously, in conjunctionwith the mass continuity equation (e.g. Armijo,1969). A comprehensive review of the mul-tiple Doppler radar analysis and the subsequentdynamic retrieval techniques can be found in theliterature (e.g. Ray, 1990).

Although dual Doppler analyses have pro-vided critical insights into understanding struc-tures of large atmospheric vortices such asTCs, dual Doppler radar analysis remains

primarily an interactive and labor-intensiveprocess with limited value for real-time app-lications. Only recently, encouraging real-timedual Doppler analysis of Atlantic hurricanes wasperformed on board the US National Oceanicand Atmospheric Administration (NOAA) WP-3D (Gamache et al., 2004). This approach haspotential, as the enhanced computer powerenables data quality control and allows navi-gation corrections to be completed in real time.Until then, practical limitations prevent muchwider applications to atmospheric vortices, espe-cially in real-time settings.

To date, multiple Doppler radar datasets ofatmospheric vortices are rare and remain mostlya privilege for specialized field campaigns.Suitable dual Doppler radar datasets areextremely difficult to collect. For large circula-tions like TCs, airborne Doppler radar datasetstypically suffer from their long sampling time(>30 minutes), where stationary assumption ofthe circulation is questionable. In addition, theaverage separation of the operational Dopplerradar networks in the United States is ∼250km.The useful dual Doppler lobes, if they exist,usually cover only a small portion of TCs’ cir-culation. For small circulations like tornadosand mesocyclones, dual Doppler datasets canonly be obtained by two or more mobileradars deployed at very close range (e.g.<5 km), such as the 3 cm wavelength (X-band)Doppler on Wheels (DOWs, Wurman et al.,1997; Wurman and Gill, 2002) and SMART-R(Biggerstaff et al., 2005).

The difficulties in sampling these stormsby multiple radars severely limit the moni-toring and study of severe weather events asso-ciated with tornados, mesocyclones, and TCsover most of the world. As a result, 2D and3D wind fields of atmospheric vortices canonly be estimated from single Doppler radarwind retrieval (SDWR) algorithms. Althoughthe kinematic structures of atmospheric vor-tices cannot be retrieved from SDWR as accu-rately and completely as from dual Doppler

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328 Wen-Chau Lee and Ben Jong-Dao Jou

radar analysis, recent advances in SDWRalgorithms have permitted the deduction ofprimary and secondary vortex structures in realtime, expanding the analyses of atmosphericvortices far beyond the restricted dual Dopplerradar domain.

Comprehensive reviews on the subject ofTCs and severe weather (tornados and meso-cyclones) from the radar perspective have beengiven by Marks (2003) and Bluestein andWakimoto (2003). This review article focuses onthe progress of SDWR algorithms, specificallytheir applications, in objectively identifying thevortex circulation center and deducing 2D and3D structures of atmospheric vortices. Section 2reviews the single Doppler radar vortex signa-tures and their interpretation. Section 3 reviewsSDWR algorithms that can deduce structuresof atmospheric vortices. Section 4 discussesobjective vortex center-finding algorithms. Thestructures of atmospheric vortices revealedfrom these SDWR algorithms are reviewed inSec. 5. Section 6 outlines how the wind fieldretrieved from the VTD algorithm can be usedwith data assimilation techniques to refine thedivergent wind estimate. Section 7 discussesthe generalized VTD technique. Section 8 offersoutlooks.

2. Interpretation of Single DopplerRadar Vortex Signatures

The single Doppler signature of a vortexis a dipole that has been recognized sincethe 1960s (e.g. Donaldson, 1970). Traditio-nally, wind fields of atmospheric vortices havebeen assumed axisymmetric and modeled by aRankine combined vortex (e.g. Rankine, 1901),characterized by two regimes: an inner coreregion with the velocity increasing linearlywith the radius (constant vorticity), and anouter potential flow region where the velocityis inversely proportional to the radius (zerovorticity) (Fig. 1).

Figure 1. Top: Single Doppler velocity signature in

a plan view of a stationary Rankine combined vortex,where the line XY is at a constant range from the radar.Bottom: Tangential velocity profile along the axis XY.Assuming a Doppler radar positioned below the bottomof the page, positive velocities are away from the radar,and negative toward (From Lemon et al., 1978.)

Starting from a Rankine combined vortex,Brown and Wood (1991) demonstrated thatDoppler velocity patterns or a ground-basedradar is modulated by (1) varying the vortexintensity and the core size, (2) adding a back-ground uniform wind field (simulating an envi-ronmental flow), (3) superimposing divergingwinds onto the Rankine combined vortex, and(4) simulating double vortices and divergencesources. Figure 2 illustrates the signatures ofa Rankine combined vortex and an axisym-metric radial flow as a function of the distancebetween the radar and the vortex, where thedipole pattern is distorted as the vortex movescloser to the radar (Wood and Brown, 1992; Leeet al., 1999). The intensity and core diameter ofthe vortex can be estimated as the magnitudeand size of the dipole. Wood and Brown (1992)showed geometrically and mathematically that(1) the dipole rotated counterclockwise when

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Figure 2. Horizontal (a) vortex and (b) divergenceflow fields. The wind vector length is proportional towind speed. The true circulation center is indicatedby the small dot at the center of each panel. Varia-tions of axisymmetric (a) vortex and (b) divergenceDoppler velocity signatures as a function of distancenorth of the radar are shown beneath the respectiveflow fields. Solid curves represent flow away from theradar (positive Doppler velocities), short dashed curvesrepresent flow toward the radar (negative Doppler velo-cities), and the thick, long-dashed curve represents flownormal to the radar viewing direction (zero Doppler velo-cities). Doppler velocity values are normalized relativeto the peak wind speed value with contour intervals of0.2. Radar is located at 100, 4, and 2 core radii south ofthe flow field (aspect ratios of 0.02, 0.5, and 1.0, respec-tively). Distances in the x and y directions have beennormalized by the core radius. In the last two panels,radar location is indicated by the large dot. (From Brownand Wood, 1991.)

a diverging wind was superimposed onto aRankine combined vortex, (2) the Dopplervelocity patterns deformed when the aspectratio, defined as the ratio of the vortex corediameter and the distance between the radarand the vortex center, increases (Fig. 2), and(3) the true vortex center and core diameter canbe derived from the distorted Doppler velocitypattern with a correction factor.

The dipole signature has been used toidentify atmospheric vortices of different scalesand interpret their structures, including themesocyclones associated with a rotating updraftcore in supercell thunderstorms (e.g. Burgess,1976), mesocyclones and tornados (e.g. Brownet al., 1978; Lemon et al., 1978), mesoscalevortices within mesoscale convective systems(e.g. Stirling and Wakimoto, 1989; Houze et al.,1989), and TCs (e.g. Baynton 1979; Wood andMarks, 1989). The Doppler velocity signaturesin real atmospheric vortices in these studies[e.g. Figs. 3(a) and 3(b)] apparently deviatefrom that of the Rankine combined vortexillustrated in Fig. 2, including: asymmetricdipole magnitude, the zero Doppler velocityisodop which does not bisect the velocity dipole,the curved and nonparallel isodop inside the coreregion, and azimuthal rotation of the dipole.Attempts have been made to recognize thesedeviations from the Rankine combined vortexradar signatures so as to infer vortex structure(e.g. Brown and Wood, 1991; Wood and Brown,1992; Lee et al., 1999).

Desrochers and Harris (1996) proposeda more sophisticated elliptical flow modelincluding both the rotational and divergentwinds to simulate the elongated Dopplervelocity patterns of the Del City, 20 May1997, OK mesocyclone. Lee et al., (1999) simu-lated Doppler velocity patterns of a Rankinevortex plus a mean flow, an axisymmetric radialflow, and tangential asymmetries from wavenumber 1 to 3. More importantly, the Dopplervelocity pattern of an asymmetric vortex variesas a function of the radar’s viewing angle

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Figure 3. The upper panels show the return power (a)and Doppler velocity (b) of the Mulhall tornado observedby a DOW. Range rings are in 1 km interval. The lowerpanels show the enlarged corresponding tornado returnpower (c) and Doppler velocity (d) structure with mul-tiple vortices (circles). (Adapted from Lee and Wurman,2005.)

(Desrochers and Harris, 1996) and/or the phaseof the asymmetric component of the vortex (Leeet al., 1999).

In summary, these Doppler velocity patternswere quite similar, and thus it is difficult to dis-tinguish their differences visually. Although qua-litative vortex structures can be inferred directlyfrom the shape and magnitudes of the dipolesignatures, deducing quantitative vortex struc-tures is not straightforward, and will rely onmore sophisticated SDWR algorithms.

3. SDWR Algorithms

Retrieval of the 2D and 3D wind fields fromsingle Doppler observations commonly followsone of two approaches: fitting the observedradial velocities to some simplified wind model,or supplementing the radar data with some

physics (e.g. conservation, “pseudo conser-vation,” momentum or vorticity equations) andseeking a flow field that satisfies the imposedconstraints (Boccippio, 1995).

The first approach typically fits the mea-sured radial velocities to a simple and specific(linear, nonlinear, or circular) model of the windfields that are representative of a weather phe-nomenon. In the radar community, the well-known approaches are the velocity azimuthdisplay (VAD) and volume velocity-processing(VVP) methods, where the analysis is radar-centered and assumes a linearly varying hori-zontal wind field in the radar sampling domain(e.g. Browning and Wexler, 1968; Caya andZawadzki, 1992; Donaldson, 1991; Matejka andSrivastava, 1991; Waldteufel and Corbin, 1979;Koscielny et al., 1982; Boccippio, 1995). Arelated approach is the velocity track display(VTD) method, which fits the Doppler velocitymeasurements to a circular model of the windfield centered at the vortex (Lee et al., 1994; Jouet al., 1994; Roux and Marks, 1996; Lee et al.,1999; Roux et al., 2004; Liou et al., 2006).

The second approach usually constrains thesingle Doppler velocity field with physical equa-tions, numerical models, or conservation ofreflectivity. These SDWR algorithms typicallyinvolve multiple volumes of radar data at con-secutive times. Assuming Lagrangian conser-vation of reflectivity, the tracking radar echoesby correlation (TREC) algorithm estimates themovement of the reflectivity field by findingthe maximum cross-correlation of features insequential radar scans or tracking echo centroids(e.g. Zawadzki, 1973; Rinehart, 1979; Symtheand Zrnic, 1983; Tuttle and Foote, 1990; Galland Tuttle, 1999). More complicated algorithmsinvolve a combination of the Lagrangian conser-vation of reflectivity and some simple assump-tions about the behavior of the radial velocityfield (e.g. Qiu and Xu, 1992; Xu et al., 1994;Shapiro et al., 1995; Liou, 1999a,b). The solutiontechniques are thus based on simple prognosticequations rather than the full set of governing

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equations to achieve lower computation costand faster processing time. Recently, Lee et al.(2003) proposed the use of the mesoscale vor-ticity method (MVM) to deduce the 3D TCstructures, including divergent wind and verticalvelocity in the inner core region, from a fre-quently observed vorticity field derived from theVTD technique.

The general impression is that favorableagreement between the wind fields retrievedfrom these SDWR algorithms and independentobservations can be achieved when the charac-teristics of the wind fields reasonably match theunderlying geometrical and/or physical assump-tions. These techniques tend not to capture thefine-scale features of the flow (i.e. higher orderand/or nonlinear features), and the uncertaintyof the circulation in the cross-beam (unmea-sured) direction is worse than in the along-beam(measured) direction. The wind fields retrievedfrom SDWR are only as good as the assumptionsused in each specific situation. In general, thealgorithms in the second approach are capableof resolving more complex wind fields thanthe algorithms in the first approach. However,when conservation of reflectivity (a commonlyused assumption in the second approach) breaksdown, such as in the inner core region of

Figure 4. Schematic showing the computation of a TREC vector to determine the motion of reflectivity echoesshaded from Time 1 to Time 2. The initial array of data at Time 1 is cross-correlated with all other second arrays ofthe same size at Time 2 whose center falls within the search area. The position of the second array with the maximumcorrelation determines the vector endpoint. (From Tuttle and Foote, 1991.)

atmospheric vortices, the algorithms in the firstapproach hold the advantage. Only those SDWRalgorithms relevant to atmospheric vortices willbe reviewed in this section.

3.1. The TREC technique

The TREC technique was first proposed byRinehart and Garvey (1978), with modifica-tions and improvements by Tuttle and Foote(1990). TREC works by storing two scans of low-elevation angle plan position indicator (PPI) orconstant altitude PPI (CAPPI) reflectivity datameasured at the same elevation angle (or at con-stant altitude) at two different times (Fig. 4).The time difference between two volumes is typi-cally a few minutes for operational radars. Thedomain at “Time 1” is divided into equal-sized2D arrays spaced some distance apart. Eachinitial array is correlated with all possible arraysof the same size at “Time 2” to find the best-matching second array with the highest corre-lation coefficient. The wind vector of that pointis determined by the distance between the firstand the second array divided by the time diffe-rences between these two scans. Such techniquesare well suited for clear air returns and con-vective storms that can produce wind estimates,

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even when the Doppler velocities are notreliable or not available (for example, beyondthe Doppler range) — a significant advantageover Doppler-velocity-based SDWR algorithms.Within the Doppler range, TREC-derived windscan be combined with Doppler velocity datato improve its quality (Gall and Tuttle, 1999).When TREC-derived winds are computed atmultiple altitudes, 3D kinematic structures canbe obtained.

Gall and Tuttle (1999) applied TREC tothree TCs — Hurricane Hugo (1989), HurricaneErin (1995), and Typhoon Herb (1996) — andobtained reasonable cyclonic circulations (e.g.Typhoon Herb in Fig. 5). TREC tends to unde-restimate the wind speed in the eyewall com-pared with the corresponding single Dopplervelocities, especially in well-organized, intenseeyewalls such as Hugo and Herb. The authorsattributed this discrepancy to the followingfactors: (1) the motion of the radar echo (pre-cipitation features) in the eyewall may be

Figure 5. Edited TREC vectors for Typhoon Herboverlaid on radar reflectivity. A gray scale key for reflec-tivity contours is shown on the upper right and a50m s−1 reference vector on the lower right. (From Galland Tuttle, 1999.)

governed by low-level convergence, and not bythe wind (i.e. violation of the intrinsic assump-tions of TREC); (2) in a vortex with high cur-vature in the wind field, it may be better forTREC to be computed in cylindrical coordinatesrather than in Cartesian coordinates; (3) rela-tively uniform reflectivity structures in the azi-muthal direction make it difficult for TREC tofind unique features to track; and (4) strongradial shear may exist across the analysis box(15 km square in their examples). The firstand third factors may be the most difficult toovercome in obtaining reasonable winds in thecritical eyewall region. Nevertheless, TREC iseffective in deducing the wind fields away fromthe eyewall of a TC, especially in determiningthe 17m s−1 wind radii.

3.2. The VTD family of techniques

The term VTD (velocity track display), coinedby Carbone and Marks (1988), refers to a displayof the NOAA WP-3D tail-mounted Dopplerradar velocities at the flight level during aradial hurricane penetration through its center[e.g. Hurricane Gloria (1985), Fig. 6]. Whena TC is axisymmetric rotation, the dipoleshould be on or parallel to the flight track.Carbone and Marks (1988) noted that thepeak Doppler velocities of the dipole in Fig. 6rotated counterclockwise from the flight track.They proposed that the azimuthal rotation ofthe dipole resulted in a radial inflow near theeyewall. The alignment of the Doppler velocitydipole relative to the flight track is affected bythe direction of the radial flow (i.e. inflow oroutflow) and the ratio of the magnitude of theradial flow to the tangential flow. Hence, the axi-symmetric tangential and radial winds can beestimated from the VTD.

The mathematical formulation of the VTDtechnique to deduce the primary circulationsof a TC was given by Lee et al. (1994).Several variations/extensions of the techniquehave been developed, namely the extended

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Figure 6. Flight-level (6.5 km) VTD display of Hurricane Gloria (1985) from the NOAA P3. The top panel showsthe Doppler velocity while the bottom panel shows the radar reflectivity factor. The Doppler velocities on the rightside of the aircraft were multiplied by −1, so the Doppler velocities do not change sign across the flight track. Theaircraft moved from east to west. (From Lee et al., 1994.)

VTD (EVTD) technique (Roux and Marks,1996), the ground-based VTD (GBVTD) tech-nique (Jou et al., 1994; Lee et al., 1999), theground-based extended VTD (GB-EVTD) tech-nique (Roux et al., 2004), and the extendedGBVTD (EGBVTD) technique (Liou et al.,2006). This subsection provides an overview ofthe VTD family of techniques using the GBVTDframework.a

The concept of the VTD technique is illus-trated in Fig. 7, where the Doppler velocity(black vectors) of an axisymmetric rotation andinflow [gray vectors in Fig. 7(a) and 7(b)] plotted

aThe GBVTD formulation is a more general form of the VTD formulation (Lee et al., 1999; Jou et al., 2008). VTDis a special case of GBVTD when a vortex is located at an infinite distance from the radar.

against the angle ψ form a negative sine andcosine curve, respectively. The Doppler velocitypattern of a combined axisymmetric rotationand radial inflow is a negative sine curve witha negative (clockwise) phase shift [Fig. 7(c)]. Itcan be shown that the sign of the phase shiftcorresponds to the direction of the axisymmetricradial flow at that radius, while the magnitudeof the phase shift is proportional to the ratio ofthe axisymmetric radial and the tangential velo-cities. By adding an environmental mean flowto the picture, the entire curve will be shiftedvertically upward or downward, depending on

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Figure 7. GBVTD concept for (a) axisymmetric rotation, (b) axisymmetric radial inflow, and (c) axisymmetricrotation and radial inflow. The left panels illustrate the circulation of axisymmetric TCs (red arrows) and theircorresponding Doppler velocities (black arrows). The right panels illustrate the contributions of Doppler velocitiesversus azimuth angle. (From Lee et al., 1999.)

the mean flow direction (i.e. the zero isodopwill not bisect the dipole in an axisymmetricvortex). In fact, this simple illustration ofthe GBVTD concept summarizes the dipolesignatures of an atmospheric vortex reviewedin Subsec. 2.2. Real atmospheric vortices alsocontain asymmetric structures; therefore, theresulting Doppler velocity profile is more com-plicated than that illustrated in Fig. 7(c). Dif-ferent wave numbers can be distinguished viaFourier decomposition of the Doppler velocityprofile along each radius in the vortex cylindricalcoordinate. Therefore, the 3D vortex structure

can be deduced by compositing GBVTD ana-lyses at each radius and height, providing thatthe circulation center at each altitude is knownaccurately. The issues related to vortex centerfinding will be discussed in Sec. 4.

Owing to insufficient independent equationsin the VTD formulation, Lee et al. (1994) andLee et al. (1999) assumed that the asymmetricradial winds are much smaller than their tan-gential counterparts, and acknowledged thatthe cross-beam mean wind component cannotbe resolved, thus letting radial winds and thecross-beam mean wind alias into tangential

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winds to close the system. Note that if theradar is located at the center of a vortex, thenonly the radial wind component can be mea-sured in that situation, and GBVTD becomesVAD. When the vortex is located at an infinitedistance (i.e. the aspect ratio is small for prac-tical purposes), the radar beams can be assumedparallel and the GBVTD formulation reduced toa simpler VTD formulation. With the knowledgeof axisymmetric tangential and radial winds ateach radius and altitude, several additional kine-matic (convergence and vertical velocity) anddynamic (angular momentum and pressure gra-dient) quantities can be computed (e.g. Leeet al., 2000; Lee and Wurman, 2005; Lee andBell, 2007).

Sensitivity tests of the GBVTD techniqueusing analytical vortices showed that accurateaxisymmetric and good asymmetric vortexstructures could be deduced from single Dopplerobservations. However, the distortion in phaseand amplitude of asymmetric vortex structuresworsens as the wave number increases (Leeet al., 1999). Note that the analytical vorticessimulated by Lee et al. (1999) did not containasymmetric radial winds; thus, the biases fromignoring asymmetric radial winds could notbe evaluated. Nevertheless, Lee et al. (1999)demonstrated that the GBVTD technique couldextract plausible kinematic and wave structuresembedded in the Doppler velocities of atmos-pheric vortices.

The EVTD technique, proposed by Rouxand Marks (1996), expanded the VTD ana-lysis to successive radial passes (legs) in orderto improve the stability of the VTD ana-lysis and extract wave number 1 radial windinformation. EVTD first solves for a constantmean wind vector by performing a VTD ana-lysis with limited coefficients on each altitude.The mean wind vector is then removed, andthe residual Doppler velocities are used tosolve for the coefficients at all radii simul-taneously by a least-squares minimization ofa cost function. This cost function minimizes

three constraints: (1) the differences betweenmean wind relative Doppler velocities andEVTD-derived radial velocities, (2) the randomvariation of coefficients between the rings, and(3) the mean wind residuals resulting from thewave number 1 asymmetry. The EVTD coef-ficients deduced in each pass are then filteredto obtain coherent mesoscale structures overtime. A similar extension of the GBVTD tech-nique, the GB-EVTD technique, was proposedby Roux et al. (2004) to process tropical cyclonedata collected by a ground-based Doppler radaron successive scans.

4. Atmospheric Vortex CirculationCenter Deduced from SingleDoppler Radar Data

Atmospheric vortex centers have different defi-nitions based on available measurements andtheir associated assumptions, including the geo-metric center, defined as the centroid of theeyewall radar reflectivity (Griffin et al., 1992);the dynamic center, defined as the minimum ofthe stream function, pressure, or geopotentialheight (e.g. Willoughby and Chelmow, 1982;Dodge et al., 1999); and the vorticity (or circu-lation) center, defined as the point that maxi-mizes the eyewall vorticity or circulation (e.g.Willoughby, 1992; Marks et al., 1992; Lee andMarks, 2000). These centers are not necessarilycollocated, and the uncertainties in estimatingthem, range from several kilometers to tens ofkilometers in TCs (Willoughby and Chelmow,1982; Lee and Marks, 2000; Harasti et al., 2004).

Uncertainties and inconsistencies of esti-mated TC centers on the order of 10 km may notaffect operational track forecasts, but they criti-cally affect computing TC wind fields in a cylin-drical coordinate system (Willoughby, 1992; Leeand Marks, 2000). This includes partitioningvortex wind fields from a Cartesian coordinatesystem to a cylindrical coordinate system (Wil-loughby, 1992; Marks et al., 1992) and retrievingvortex wind fields from single Doppler radar

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data directly in cylindrical coordinates (Leeet al., 1994; Roux and Marks, 1996; Lee et al.,2000; Roux et al., 2004; Lee and Wurman, 2005).Lee and Marks (2000) demonstrated that avortex circulation center needs to be accuratewithin 1 km on a 20 km radius of maximumwind (RMW), or 5% of the RMW, to keep thenominal error of the apparent wave number 1less than 20% of the maximum axisymmetrictangential winds (MATWs: VT0−max) in theGBVTD analysis. Since the vorticity (circu-lation) center is directly related to the accuracyof the deduced vortex wind fields, it will be thefocus of this section.

Harasti and List (2005) applied a principalcomponent analysis (PCA) to the single Dopplervelocity data arranged sequentially in a matrixaccording to the range and azimuth coordinates.For a Rankine vortex, the coordinates (location)of particular cusps in the curve of the firsttwo eigenvector coefficients plotted against theirindices are geometrically related to both the cir-culation center and the RMW. The uncertaintyof the PCA method regarding asymmetric TCsis still a research topic.

The vorticity center can be computed froma given 2D horizontal vortex wind field and asimplex method (Neldar and Mead, 1965) toidentify a point that maximizes the eyewall vor-ticity (Marks et al., 1992). However, estimatingTC centers using this technique is not straight-forward with a single Doppler radar, becausethe velocity patterns do not provide a com-plete 2D wind field for direct vorticity compu-tation. Lee and Marks (2000) proposed a novelapproach, and invented the GBVTD-simplexalgorithm, by using the simplex method to ite-ratively identify a center that maximized theGBVTD-derived eyewall vorticity. The searchprocess of the simplex algorithm toward the truecenter of a Rankine vortex centered at (0, 60),with an RMW of 20 km and a maximum tan-gential wind of 50m s−1, is illustrated in Fig. 8.

The simplex method uses three operations —reflection, expansion, and contraction — based

Figure 8. Illustration of the GBVTD-simplex conver-gence (thick dashed line) and the intermediate simplexfrom the initial guess at I (−10.0, 50.0). The triangleA–B–C is the initial simplex with a centroid at I. R,X, and C indicate the resulting points from reflection,expansion, and contraction operations with the numbercorresponding to the individual simplex. The TC centeris located at (0.0, 60.0) with a hurricane symbol. Onlythe first six iterations are shown here. The GBVTD-derived axisymmetric tangential winds are listed in par-entheses in m s−1. Differently shaped lines distinguishintermediate triangles. (From Lee and Marks, 2000.)

on the value of interest to determine thesubsequent search toward the maximum orminimum value. In this example, the MATWsare computed starting from vertices A, B, andC as the vortex centers of GBVTD analyses.Vertex A has the lowest MATW value, yielding areflection point of A (R1) on segment BC. Thenthe MATW using R1 as a center [MATW(R1)]is computed and compared with those MATWsin A, B, and C. Subsequent operations todetermine a new vertex replacing A to form thenext simplex are based on whether MATW(R1)is a maximum (expansion), intermediate (R1replaces A), or minimum (contraction). Theabove processes are repeated until a convergencecriterion is reached. The thick, dashed line (inFig. 8) represents the converging path of inter-mediate TC centers toward the true center at(0, 60).

The simplex method described here is effi-cient in finding a local maximum or minimum

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within the parameters specified by the users. Inpractice, it is recommended to run the simplexalgorithm from various initial guesses, pre-ferably from all quadrants of the “true” center.Without independent verifications, the groupingor scattering of the simplex results (end points)is the only statistical indication of the quality(uncertainty) of the GBVTD-simplex-derivedvortex center. The mean error in the GBVTD-simplex-derived vortex centers using analyticalvortices is 0.35m (< 2% of a 20 km RMW),which is significantly less than the 1 km radialgrid spacing in the cylindrical coordinates.Random errors of 5m s−1 added to the Dopplervelocity do not change the mean error andstandard deviation much. When applied to realatmospheric vortices, the standard deviationsincreased to ∼2 km in Typhoon Alex (∼9% of a23 km RMW; Lee and Marks, 2000) and ∼30min the Mulhall tornado (∼5% of a 600m RMW;Lee and Wurman, 2005). The accuracy of theGBVTD-simplex algorithm strongly depends onthe accuracy of the GBVTD-derived MATW.Since the MATW can be biased by the unknowncross-beam mean flow and the unresolved wavenumber 2 radial flow, the GBVTD-simplex algo-rithm has experienced difficulties in deducingcenters in elliptical (wave number 2) vorticessuch as Typhoon Herb and Nari (Michael Bell,2005; personal communication). Further quan-tifying the error characteristics of the GBVTD-simplex algorithm in real atmospheric vorticesrequires independent measurement of the “true”vortex center from either in situ measurementsor a second set of Doppler radar observationsfrom a different viewing angle.

5. Atmospheric Vortex Structures

5.1. Structure of tropical cyclones

The first physically consistent 3D axisymmetrickinematic structure of a TC (Hurricane Gloria,1985) deduced from NOAA WP-3D singleDoppler radar data and the VTD technique wasgiven by Lee et al. (1994). Figure 9 illustrates

axisymmetric structures of a TC previously onlyavailable from flight-level composite data (e.g.Jorgensen, 1984a,b), pseudo-dual Doppler ana-lysis (e.g. Marks and Houze, 1987; Marks et al.,1992; Franklin et al., 1993), or numerical simu-lations (e.g. Rotunno and Emanuel, 1987). Keyfeatures in Fig. 9 include:

(1) An upright eyewall reflectivity (∼R =13km) is accompanied by an MATWexceeding 60m s−1 at an RMW of ∼18 km,while the axisymmetric tangential windsdecrease rapidly both inside and outsidethe RMW [Fig. 9(a)]. The decrease ofthe MATW with height suggests a warmcore structure from the thermal windrelationship.

(2) A shallow, low-level inflow (∼1 km deep)inside R ∼ 32km peaks (> 3 m s−1) as itapproaches the eyewall (Fig. 9b). It con-verges near the eyewall [Fig. 9(c)] and turnsinto an updraft in the eyewall [> 4m s−1;Fig. 9(d)].

(3) Immediately outside the eyewall, a region ofdowndraft (> 2m s−1, between R = 16kmand 40km) is consistent with weak reflec-tivity and radial inflow beneath the anviloutflow [dashed lines between 8 km and13 altitude in Fig. 9(b)]. This branch ofinflow beneath the anvil was seldom resolvedin previous pseudo-Doppler analysis butappeared in numerically simulated TCs[Fig. 5(c) in Rotunno and Emanuel, 1987].

The VTD-derived asymmetric structure ofGloria over two successive legs suggested thatthe hurricane evolved from a wave number 1[Fig. 10(a)] to a wave number 2 [Fig. 10(b)]structure within an hour. A pseudo-dualDoppler (PDD) analysis using combined datafrom these two legs showed a wave number3 structure [Fig. 10(c)], suggesting temporalaliasing in the pseudo-dual Doppler analysis.Again, the advantage of reducing the effectivestationary time period from ∼1 hour (PDD) to∼20 minutes (VTD) was clearly demonstrated.

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Figure 9. Radius–height cross-sections of (a) the mean tangential velocity, (b) the mean radial velocity, (c) themean divergence, and (d) the mean vertical velocity, with a superimposed reflectivity region in shading. Positive valuedcontours represented by thin, solid lines; negative-valued lines are thin and dashed. A zero contour is represented bya thick, solid line. Units and the contour interval are indicated at the top of each panel. The cross-section extendsfrom the hurricane center. (From Lee et al., 1994.)

Lee et al. (1994) derived a hodograph of theEarth-relative horizontal wind from these twolegs of data (Fig. 11) and found that Gloria’smean motion vector is ∼30◦ to the left of thedeep-layer, density-weighted mean wind vector,consistent with those computed from the deep-layer mean wind in a much larger domainof several degrees in longitude and latitude(e.g. George and Gray, 1976; Chan and Gray,1982; Holland, 1984; Dong and Neumann, 1986;Hanley et al., 2001).

The structure and evolution of HurricaneHugo (1989) over a 7-hour period on 17 Sep-tember 1989 were presented by Roux and Marks(1996) using EVTD-derived winds from six suc-cessive eye crossings about 70 minutes apart.The axisymmetric and asymmetric structures inHugo were qualitatively consistent with those

in Gloria, Alicia, and Norbert. More import-antly, Roux and Marks (1996) demonstratedthat plausible wave number 1 radial wind couldbe derived from the EVTD technique using mul-tiple legs of airborne Doppler radar data. TheEVTD-derived radial wind rotated from easterlyat a 2 km altitude to southerly at 5 km andbecame southwesterly at 10 km, and comparedfavorably with the PDD-derived radial winds(Fig. 12).

The ability to deduce kinematic structuresof landfalling TCs from coastal Doppler radardata has been demonstrated through TyphoonAlex (1987) (Lee et al., 2000), the concentriceyewall (double wind maxima) in TyphoonBillis (2000) (Jou et al., 2001), Hurricane Bret(1999) (Harasti et al., 2003), and HurricaneCharley (2004) (Lee and Bell, 2007) using the

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Figure 10. Total tangential winds (contours) andreflectivity (gray shades) for (a) the east–west leg, (b) thenorth–south leg, and (c) the PDD analysis at a 3.5 kmaltitude. (Adapted from Lee et al., 1994.)

Figure 11. Hodograph of the Earth-relative horizontalwinds and its deep-layer mean obtained from the VTDanalysis (VM ) and the 12-hour mean motion of HurricaneGloria (1985), VH . (From Lee et al., 1994.)

GBVTD analysis, Typhoon Nari (2001) usingthe EGBVTD technique (Liou et al., 2006),and Cyclone Tina (2002) (Roux et al., 2004)using the GB-EVTD technique. The structuresof these landfalling TCs were consistent withprevious studies but revealed detailed evo-lution of TCs (∼6–15minute interval) comparedwith those deduced from airborne dual Dopplerradar data (∼30–60minute interval). Typicaltyphoon inner core structures are illustrated inFigs. 13(a) and 13(b). Lee et al. (2000) tookthe axisymmetric structure a step further tocompute angular momentum and cyclostropicperturbation pressure in Alex (Fig. 13). Theangular momentum pattern [Fig. 13(c)] isnear-upright in the eyewall, similar to thosededuced from dual Doppler analysis in Hur-ricane Norbert [Fig. 12(f) in Marks et al. 1992].The retrieved perturbation surface pressure atthe center [Fig. 13(d)] is 23 hPa lower thanthe pressure at 60 km radius. The radial per-turbation pressure pattern compared favorablywith the pressure pattern derived from surfacepressure measurements assuming axisymmetric

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Figure 12. Horizontal cross-sections of the radial windcomponent at (a, b) 2 km, (c, d) 6 km, (e, f) 10 kmaltitudes. The left panels (a, c, e) refer to EVTD-derivedvalues, the right panels (b, d, f) to PDD-derived values.(From Roux and Marks, 1996.)

pressure distribution from R = 20km to R =60km (Fig. 14; note that there is no surfacepressure measurement within 20 km of the TCcenter). This agreement suggests good internalconsistency of the GBVTD-retrieved axisym-metric circulation and the retrieved pressuredeficit, which is likely accurate to ∼2–3hPa ifthere is no significant aliasing from the unre-solved cross-beam mean wind and wave number2 asymmetric radial wind. Lee and Bell (2007)showed that the rapid intensification process ofHurricane Charley (2004) could be captured byGBVTD-retrieved pressure deficit in real time.The result is very encouraging, and it is feasibleto estimate the central pressure of landfalling

TCs from coastal Doppler radar data. This couldbe a valuable tool for diagnosing the intensity oflandfalling TCs.

5.2. Structure of mesoscale vortices

in MCS

Zhao et al. (2007) presented the first GBVTD-derived 3D kinematic structure of a mesovortexembedded within a quasi-linear convectivesystem (QLCS) over the northern Taiwan Strait.A hook-echo-like structure [Fig. 15(a)] deve-loped at the southern end of a convective linesegment and possessed a pronounced velocitydipole signature. The asymmetric structure ofthe mesovortex at 4 km [Fig. 15(a)] indicatesthat the maximum wind of 25m s−1 was locatedin the rear of the mesovortex (upper leftcorner). The asymmetric component of the tan-gential wind was retrieved to wave number2 and most of the amplitude was containedin the wave number 1 component. At a 2 kmheight, the kinematic structure of the vortex[Fig. 15(b)] was quite different. In the rear ofthe mesoscale vortex, there was a secondarymaximum of tangential wind consistent with thewind distribution at 4 km. However, the mostintense tangential wind of 25m s−1 was locatedat the front quadrant of the mesoscale vortex(upper right). Through examining the sequenceof the Doppler radial velocity, it is evident astrong rear-to-front flow developed during theformation period of the mesoscale vortex. Thedownward penetration of this rear-to-front jetenhanced the low-level cyclonic vorticity. At theleading edge of this rear-to-front jet there wasa pronounced flow convergence triggering a lineof convection evident in the reflectivity field atthe 2 km height.

The MATW [Fig. 15(c)] at the lowestretrieved level (1 km height) is 18m s−1. It isconsistent with the location of the maximummean reflectivity field. The axis of the MATWis tilted inward to the center of the vortex with

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Figure 13. Axisymmetric structure of (a) tangential winds, (b) radial winds, (c) angular momentum, and (d)perturbation pressure, at 0417 LST. Solid (dashed) lines represent positive (negative) values. The thick, solid line isthe zero contour. Axisymmetric reflectivity factors are in color. Vectors in (b) are composed of axisymmetric radialwinds and vertical velocities. (Adapted from Lee et al., 2000.)

increasing altitude. This feature stands in con-trast to the inner core structure of a maturetropical cyclone in which the tilt is outwardwith increasing altitude. The mean radial wind[Fig. 15(d)] outside (inside) the RMW is inward(outward) with an intensity of 5m s−1 (2 m s−1)and convergence near the RMW. It has beendemonstrated in the above presentation thatthe GBVTD technique is a powerful tool forretrieving the kinematic structure of mesoscalevortices.

5.3. Structure of tornados

The first GBVTD analysis of a tornado,near Bassett, Nebraska, was presented by

Bluestein et al. (2003), who illustrated coherenttornado structures over a 3.5-minute period.The W-band radar (Bluestein et al., 1995) canonly scan in one elevation angle; therefore, onlyhorizontal structure can be deduced from thisdataset. The Doppler velocity pattern and thecorresponding GBVTD analysis of the Bassetttornado are deduced (Fig. 9 in Bluestein et al.,1995). However, with a scan update time ofapproximately 12 seconds, unprecedented evo-lution of the tornado was revealed. The axi-symmetric tangential and radial winds overthe 3.5minutes are portrayed in Fig. 16. TheMATW is ∼28m s−1 at R ∼ 170m. The axi-symmetric radial winds are positive (outflow)inside R = 200m and negative (inflow) outside

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Figure 14. (a) The surface stations report hourly datain northern Taiwan. Typhoon locations at 0502 and 0602LST are indicated by the typhoon symbols. The terrainis in gray shades. (b) The thick lines are the GBVTD-derived axisymmetric perturbation pressure (P ′) at thesurface. Circles are the perturbation pressures computedfrom five surface stations. (Adapted from Lee et al.,2000.)

R = 200m which suggests a two-cell circu-lation. The switch-over point of the inflow andoutflow is outside the RMW. This may be anartifact of the additional centrifugal effects expe-rienced by the debris (targets of the radar),not by the winds (Dowell et al., 2005). Thestructure of another tornado, near Stockton,

Kansas, collected by the same radar, has beenpresented by Tanamachi et al. (2006). The near-stationary wave number 2 structure was attri-buted to the superimposition of an axisymmetricvortex onto a near-stationary deformation field(Bluestein et al., 2003). Recent studies by Leeet al. (2006a) and Tanamachi et al. (2006) indi-cated that this near-stationary wave number2 asymmetry might be an artifact due to thespatial aliasing of a fast-moving tornado and arelatively slow scan rate of the W-band radar inthis particular situation. Research is underwayto systematically examine this issue.

In contrast, DOWs are capable of scanningmultiple elevation angles at a slower volumeupdate rate (∼1 minute), but the 3D structuresof tornados can be retrieved. Lee and Wurman(2005) produced the first comprehensive studyof tornado circulations collected by DOWs on4 May 1999 near Mulhall, Oklahoma, USA.The axisymmetric structure of the Mulhalltornado at 0310:03 (hhmm:ss) UTC is pre-sented in Fig. 17. The axisymmetric tangentialwind [Fig. 17(a)] (hereafter, all quantities areaxisymmetric unless stated otherwise) profilesresemble a miniature, intense TC (about 1/15in length scale) and possess characteristics com-monly seen in the inner core region of a matureTC (e.g. Marks et al., 1992; Lee et al., 1994;Roux and Marks, 1996). The depth of theinflow layer reached 1 km but the most intenseinflow was clearly confined near the surface.The downdraft magnitude of 30m s−1 is com-parable to the estimated downdraft speed inthe Dimmitt tornado with similar intensity(Wurman and Gill, 2000). The secondary (meri-dional) circulation [Fig. 17(b)] provided the firstobservational evidence that a classical two-cellcirculation, commonly seen in tornado vortexchamber and numerical simulations with a swirlratio greater than 1 (e.g. Ward, 1972; Rotunno,1979), does exist in intense tornados. Theswirl ratios of the Mulhall tornado computedfrom the GBVTD-derived axisymmetric circu-lation during the entire observational period

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Figure 15. GBVTD-derived horizontal vortex circulations at (a) 2 km and (b) 4 km altitudes. The lower panels

show (c) the axisymmetric tangential winds and (d) radial winds of the mesocyclone.

(∼14 minutes) are all above 2 (Table 1),consistent with the two-cell structure in thesecondary circulation and the multiple vorticesobserved in the Mulhall tornado. A strikingexample is seen in Figs. 3(c) and 3(d), where sixsmall vortices can be identified on the west sideof the tornado, especially in the velocity field,

at 1316:28 (Wurman, 2002; Lee and Wurman,2005).

The pressure deficit from advection terms[Fig. 17(c)] is consistent with the secondary cir-culation, a component rarely, if ever, resolvedin past observational studies of tornadoes. Thetotal pressure deficit [Fig. 17(g)] at z = 50m

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Figure 16. Averaged axisymmetric tangential andradial winds obtained by averaging across all radii. (FromBluestein et al., 2003.)

is 81 hPa lower than the pressure at R =3km, which is dominated by the cyclostrophicpressure [Fig. 17(e)] with adjustment from theadvection pressure [Fig. 17(c)]. These pressuredeficits are comparable with rare in situ obser-vations (Winn et al., 1999; Lee et al., 2004;Wurman and Samaras, 2004; Lee et al., 2004)in strong tornadoes. This magnitude is also con-sistent with analytical pressure analysis froma similar wind profile of a Rankine combinedvortex (see Appendix in Lee and Wurman,2005).

The vorticity pattern in Fig. 17(f) showsan annular (or so-called “ring”) vorticity profilewhere the peak vorticity of 0.28 s−1 is concen-trated in an annulus between 300 and 500min radii. The radial vorticity gradient of a ringvorticity profile changes sign and satisfies thenecessary condition for barotropic instability,usually accompanying mature TCs (Mallenet al., 2005). The angular momentum contours[Fig. 17(h)] nearly upright inside the RMW andtheir value increase with the radius while theangular momentum outside the RMW increasedat a slower rate. This pattern is quite similar tothose resolved within a mature TC and suggeststhat the low-level inflow brings in higher angular

momentum and the secondary circulation main-tains the vortex (e.g. Lee et al., 2000; Markset al., 1992; Rotunno and Emanuel, 1987).

6. Assimilating VTD-DerivedWinds into the Numerical Model

The strength of the VTD family of SDWR tech-niques is in providing good axisymmetric andasymmetric tangential winds but lacking asym-metric radial winds. One of the outstandingissues is whether the asymmetric divergent com-ponent can be extracted from the VTD-derivedincomplete wind fields in order to initializenumerical simulations or data assimilation.

Lee et al. (2003) proposed an innovativeconcept using the mesoscale vorticity method(MVM) to retrieve inner core divergence andvertical velocity from the frequently availablevertical vorticity field from MM5 simulations.The MVM derives vertical velocity from vor-ticity variation in space and time based onthe mesoscale vorticity equation, obtained byneglecting the solenoidal term in the full equa-tions. When the mesoscale vorticity equation isformulated in Lagrangian coordinates composedof three nonlinear ordinary differential equa-tions, the vertical velocity can be solved nume-rically along each characteristic with properboundary conditions. In their study, observingsystem simulation experiments were under-taken to demonstrate that the vertical velocitiesderived from the MVM were less susceptible toerrors in horizontal winds than the kinematicmethod for TCs.

Lee et al. (2006) applied the MVM toderive the inner core vertical velocity anddivergent component of the horizontal windfrom single Doppler radar observations ofHurricane Danny (1997) using three conse-cutive volumes of GBVTD analysis. The MVM-retrieved divergent component of the windvectors superimposed on radar reflectivity ofthe inner core region of Danny is illustrated inFig. 18(a). A major convergence zone was found

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Table 1. Swirl ratio (S) of the Mulhall tornado from0310:03 to 0323:12 and the key parameters used in thecalculation. See details in Lee and Wurman (2005).

UTC R km h km ur m/s vr m/s S

0310:03 2 1.0 16 35 20310:57 1.5 0.5 16 40 40312:05 1.5 0.6 15 40 30310:59 1.5 0.5 10 40 60314:16 2.0 0.8 12 40 40315:13 2.5 0.6 16 30 40316:28 1.5 0.6 16 40 30317:22 1.5 0.6 8 32 50318:25 2.5 0.8 16 25 20319:18 2.8 0.7 20 20 20320:24 3.0 0.7 20 25 30321:20 3.0 1.0 18 25 20322:16 3.0 1.0 10 20 30323:12 3.0 1.0 12 20 3

at the RMW, east of the eyewall over the areaof a bow-shaped echo. The updraft inside theecho drew in significant air from the surroundingarea, including air from the eye, to sustainthe convection. Another major convergence areaassociated with high reflectivity was located inthe northwest of the eye. Vertical structure por-trayed on an east–west cross-section through theeye is shown in Fig. 18(b). The wind vectorsare composed of the density-weighted divergentE–W component and the vertical velocity. Theprominent feature is the narrow band of updraftlocated 20 km east of the eye, where flow con-verges from the east and west of the line.The outward tilt of the updraft matches thereflectivity pattern well. The strength of theupdraft increases with height and peaks atabout a 6–7km altitude, above which the flowdiverges away from the eyewall. Also shown isa secondary circulation induced by the eyewallupdraft including low-level convergence, upper-level divergence, and weak downward motioninside the eye (x = 0km).

The initial success of this approach notonly provides an avenue for retrieving asym-metric radial winds from the VTD ana-lysis, but also opens the door to use theGBVTD-MVM-derived, dynamically consistent

vortex structures as initial conditions for simu-lating atmospheric vortices. The update rate of6minutes will enable the use of these dynami-cally consistent wind fields in data assimilation.This is an area that still needs to be explored infuture research.

7. The Generalized VTD (GVTD)Technique — A New Paradigm

Since the invention of weather Doppler radar,the kinematic structures of weather phenomenahave been displayed exclusively in radial velocity(Vd) in a spherical coordinate system. The cha-racteristics and limitations of atmospheric vor-tices displayed in Vd have been discussed indetail in this article. A new paradigm was pro-posed by Jou et al. (1996) to display the atmo-spheric vortex in VdD/RT space rather than inthe convectional Vd space. Its potential impacton atmospheric vortex research was not rea-lized until recently. Displaying the atmosphericvortex in VdD/RT space eliminates the geo-metric distortion of a vortex inherent in thetraditional Vd space. In this framework, thedipole structure of an axisymmetric vortex isa function of the linear azimuth angle and isnot distorted by a varying aspect ratio. The-refore, the center will always be the half pointon the line connecting the peak inbound andoutbound VdD/RT (Jou and Deng, 1997). Jouet al. (1996) named this center-finding algorithmthe velocity distance azimuth display (VDAD)method.

The mathematic foundation of representingatmospheric vortices in VdD/RT space andthe characteristics of the VDAD method havebeen documented and examined by Jou et al.(2008), who demonstrated that the GBVTD for-mulation when representing in VdD/RT spacebecomes exact mathematically without the needto approximate cosα as required in GBVTD.This new approach is named the generalizedVTD (GVTD) technique. Hence, the geometricdistortion of the dipole inherent in Vd space

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Figure 17. Axisymmetric structure (radius–height) of the Mulhall tornado at 4 May 1999 0310:03 UTC. The returnpower is shown in gray shades and contours represent (a) tangential wind, (b) radial wind, (c) advection pressure deficit,(d) divergence, (e) cyclostrophic pressure deficit, (f) vorticity, (g) total pressure deficit, and (h) angular momentum.The vectors in (b) illustrate the secondary circulation of the Mulhall tornado. Solid (dashed) lines represent positive(negative) values. (From Lee and Wurman, 2005.)

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Figure 18. Radial flows superimposed on the radar echoes over the area surrounding the bow-shaped and sausage-shaped echoes. (b) Vertical structure of the secondary circulation wind vectors composed of the east–west divergentand the vertical velocities. (From Lee et al., 2005.)

and the aliasing at higher wave numbers in theGBVTD-retrieved vortex circulation are effec-tively removed in the GVTD technique. Moreimportantly, the constant mean wind appears asa set of parallel lines in VdD/RT space, a muchsimpler signature to identify than the diverginglines in Vd space. Jou et al. (2008) also showed

that VTD and GBVTD are special cases ofGVTD. A comparison of the differences betweenthe GBVTD- and GVTD-derived vortex struc-tures from wave number 0 to 3 is illustratedin Fig. 19. Clear advantages of the GVTDtechnique can be seen in the retrieved vortexstructure, especially in wave numbers 2 and 3.

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Figure 19. Comparison of GBVTD- and GVTD-retrieved vortex structure. (a1) The simulated axis-symmetry windfield, (a2) the GBVTD-retrieved wind field, and (a3) the GVTD-retrieved wind field. (b1–b3) The same as (a1–a3)but for the wave number 1 case, (c1–c3) the same as (a1–a3) but for the wave number 2 case; and (d1–d3) the sameas (a1–a3) but for the wave number 3 case. (From Jou et al. 2008.)

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GVTD also expands the analysis domainto the point where it is no longer limitedby R/RT < 1 as in GBVTD. The dramaticimprovement occurs when R/RT > 1, i.e.the radar is located inside the RMW whereit does not observe the full component of themaximum tangential wind of a vortex. Whilethe GBVTD technique cannot retrieve the fullcirculation in this situation, the GVTD tech-nique can still retrieve the full tangential windcomponent (Fig. 20). Therefore, the GVTDtechnique can significantly expand the analysisdomain. However, it still suffers from the VTDclosure assumption, where the asymmetric tan-gential and radial winds cannot be separated inits current framework.

8. Outlook

SDWR algorithms remain the only practicalavenue for deducing 3D atmospheric vortexstructures for research and operation forecasts inthe foreseeable future. While there are obviouslimitations associated with the VTD family oftechniques discussed in this article, it is evidentthat the VTD family of SDWR algorithms canprovide a more complete picture of atmosphericvortices than other SDWR techniques. Whilethis review article focuses on research resultsthat have already appeared in refereed journals,research efforts to address these challenges arepresented in this section.

8.1. Improve mean wind estimates

The GBVTD primary circulation can be biasedby the cross-beam component of the environ-mental wind. This bias ultimately affects esti-mates of the central pressure (Lee et al., 2000;Lee and Bell, 2004). Initial attempts indicatedthat this component of the environmental windcould be estimated by the hurricane volumevelocity processing (HVVP) method (Harasti,2003, 2004; Harasti and List, 1995; Harasti andList, 2001) and the GVTD technique. Harastiet al. (2005) included the mean wind derived

from HVVP and produced 3–8hPa correctionsto the GBVTD pressure values in HurricaneCharley (2004). The special signature of themean wind in the VdD/RT space provides apromising way to accurately estimate the meanwind (Jou et al., 2008). An objective method toestimate the mean wind vector is under deve-lopment. Future research will need to addressthe quality of the mean winds derived from thesetechniques and to validate the pressure retrievalresults for other TCs with dropsonde data andfor tornados with in situ measurements.

8.2. Improving the VTD closure

assumptions

The VTD closure assumptions, based on thelevel of understanding of TC structures anddynamics in the early 1990s, can certainlybe improved. It has been shown that vor-ticity waves [so-called vortex Rossby waves(VRWs); Montgomery and Kallenbach, (1997)]live and propagate along vorticity gradientsnear the inner core of a TC. Their asso-ciated potential vorticity mixing processes havebeen widely accepted as the mechanism forforming polygonal eyewalls, mesovortices, andspiral rainbands (e.g. Guinn and Schubert, 1993;Montgomery and Kallenbach, 1997; Schubertet al., 1999; Kossin et al., 2000). The exis-tence of the VRW in hurricane-like vortices hasbeen simulated in numerical models of varyingcomplexity, ranging from a barotropic, nondi-vergent, three-region vortex model (e.g. Terweyand Montgomery, 2002) to full-physics, nonhy-drostatic models (e.g. Wang, 2002a, b; Chen andYau, 2001).

The VRW dynamics provides phase andamplitude relationships between asymmetrictangential wind and radial wind that may beused as closure assumptions for the VTD for-mulation. A simple form of the VRW is anidealized, small-amplitude, vortex edge wavetraveled on the vorticity gradient of a Rankinevortex (Lamb, 1932) where a wave number 2

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Figure 20. Comparison of Vd and VdD/RT displays for two different radii (delineated as red and blue lines) fortwo vortices with different RMWs: (a) Vd display for a pure rotating vortex with Rmax = 30 km; (b) Vd profiles atR = 30 and 60 km; (c) same as (a) except for the VdD/RT display; and (d) same as (b) except for the VdD/RT

profiles; (e)–(h) are the same as (a)–(d) except for Rmax = 80 km > RT and the two Vd profiles are at R = 80 and110 km. The center, T, is located at (x, y) = (200 km, 200 km) and the hypothetical Doppler radar, O, is located at(x, y) = (150 km, 150 km) with RT = 50

√2 km. (From Jou et al. 2008.)

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disturbance possesses an elliptic streamline (e.g.Lee et al., 2006a). Lee et al. (2006a) also showedthat a deformation field is essentially a wavenumber 2 disturbance in cylindrical coordinates.They demonstrated that the GBVTD-retrievedsignatures are different between these two typesof elliptical vortices. Lee et al. (2002) applied thesimple vortex edge wave model in the GBVTDanalysis of Typhoon Herb (1996) and resolvedtwo pairs of counter rotating vortices propa-gating downstream, consistent with the coun-terclockwise rotation of the elliptical reflectivitypattern. However, the sharp vorticity gradientof the Rankine vortex created an unrealistic dis-continuity near the RMW. One of the challengesis how to formulate the VRW disturbance in rea-listic axisymmetric, atmospheric vortices wherethe simple form of the equations illustrated byLamb (1932) is no longer valid.

8.3. Quantifying uncertainties

of the GBVTD-simplex

center-finding algorithms

Efforts have been made to quantify the uncer-tainties of the GBVTD-simplex algorithm.Murillo et al. (2001) statistically evaluated thecenter information (including location, RMW,and MATW) over time from independentlyderived centers from two separate WSR-88Dsin Hurricane Danny (1997). They found thatthe individual tracks computed from two radarsare consistent but that variations of indi-vidual centers can be large. A more consistentset of centers can be obtained by simulta-neously considering time continuity in centerlocations RMW and MATW (Murillo et al.,2002). Bell and Lee (2002, 2003) incorporatedstatistical information on locations, RMWs,and MATWs of past centers to objectivelyselect the most probable center at each time,and the procedure can be automated after 6–7 time periods, when reliable statistics canbe built. Future research needs to addressthe uncertainties in the GBVTD-simplex algo-rithm resulting from multiple wind maxima (e.g.

concentric or nonconcentric eyewalls) and thebiases in the axisymmetric tangential windsfrom wave number 2 radial winds.

8.4. Operational aspects

Algorithms using the vortex reflectivity anddipole Doppler velocity structure to identifythe vortex center and maximum wind speedhave been implemented for many years (e.g.Lemon et al., 1978; Wood and Brown, 1992;Wood, 1994). The EVTD-derived TC innercore structure in Hurricane Olivia (1994) wastransmitted to the Tropical Prediction Center(TPC) via satellite link (Dodge et al., 1995).More complicated suites of algorithms thatcombine these simple algorithms with severaladvanced techniques described in this reviewhave drawn attention in recent years from ope-rational centers such as the Central WeatherBureau in Taiwan (P.-L. Chang, personal com-munication) and the TPC (e.g. McAdie et al.,2001; Harasti et al., 2004). Digital radar dataare available at operational centers, openingup the possibility of applying algorithms dis-cussed in this article in operational settings inreal time.

The ability to estimate the absolute centralpressure and capture the rapid intensificationin Hurricane Charley was attempted in Harastiet al. (2005) where the GBVTD-HVVP-derivedpressure fields were within 2–3hPa of the centralpressure measured by the GPS dropwindsondes.A software package, Vortex Objective RadarTracking and Circulation (VORTRAC), basedon the GBVTD-HVVP framework, has beendeveloped and transferred to the TPC and field-tested in the 2007 hurricane season (Lee et al.,2006b; Harasti et al., 2007). VORTRAC hasbeen officially accepted by the TPC/NHC foroperation use starting from the 2008 hurricaneseason.

Adequate scanning strategy selected byradar operators is essential for the success ofretrieving TC wind fields using the VTD familyof SDWR techniques. Radar operators have

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to choose the tradeoff between range folding(covering more storms) and velocity folding(detecting maximum wind). Examples shownby Lee and Bell (2004) suggested that it ismore important to choose a pulse repetition fre-quency that gives the maximum range whensampling TCs because velocity folding is gene-rally recoverable.

Further research aimed at integrating andcapitalizing on the strengths of the individualalgorithms described in this article, along withimproved physical understanding of the under-lying vortex dynamics, will help to betterconstrain the vortex solutions obtained fromsingle Doppler radar data. These dynamicallyconsistent wind fields can then be used foroperational guidance, as initial conditions ordata assimilations in numerical models, and asresearch tools for gaining further insight intoatmospheric vortex structure.

Acknowledgements

The authors thank Mr M. Bell, Mr S. Ellis,and the two anonymous reviewers for theirthoughtful comments that drastically improvedthis article.

[Received 26 July 2007; Revised 26 August 2007;Accepted 30 September 2007.]

References

Armijo, L., 1969: A theory for the determinationof wind and precipitation velocities with Dopplerradars. J. Atmos. Sci., 26, 570–573.

Baynton, H. W., 1979: The case for Doppler radarsalong our hurricane affected coasts. Bull. Amer.Meteor. Soc., 60, 1014–1023.

Bell, M., and W.-C. Lee, 2002: An objective methodto select a consistent set of tropical cyclone circu-lation centers derived from the GBVTD-simplexalgorithm. Preprints, 25th Conf. on Hurricaneand Tropical Meteorology, (29 Apr. 1–3 May2002, San Diego, CA, USA, Amer. Meteor. Soc.),pp. 642–643.

Bell, M., and W.-C. Lee, 2003: Improved tro-pical cyclone circulation centers derived from

the GBVTD-simplex algorithm. Preprints, 31stConf. on Radar Meteorology, (Seattle,Washington, USA, AMS), pp. 1012–1014.

Biggerstaff, M. I., L. J. Wicker, J. Guynes,C. Ziegler, J. M. Straka, E. N. Rasmussen,A. Doggett, IV, L. D. Carey, J. L. Schroeder,and C. Weiss, 2005: The shared mobile atmo-spheric research and teaching radar: a collabo-ration to enhance research and teaching. Bull.Amer. Meteor. Soc., 86, 1263–1274.

Bluestein, H. B., and R. M. Wakimoto, 2003: Mobileradar observations of severe convective storms.In Radar and Atmospheric Science: A Collectionof Essays in Honor of David Atlas, eds. R.Wakimoto and R. Srivastava (Amer. Meteor.Soc.), pp. 105–136.

Bluestein, H. B., A. L. Pazmany, J. C. Galloway, andR. E. McIntosh, 1995: Studies of the substructureof severe convective storms using a mobile 3-mm-wavelength Doppler radar. Bull. Amer. Meteor.Soc., 76, 2155–2169.

Bluestein, H. B., W.-C. Lee, M. Bell, C. Weiss, andA. Pazmany, 2003: Mobile-Doppler-radar obser-vations of a tornado in a supercell near Bassett,Nebraska on 5 June 1999. Part II: Tornado-vortexstructure. Mon. Wea. Rev., 131, 2968–2984.

Boccippio, D. J., 1995: A diagnostic analysisof the VVP single-Doppler retrieval technique.J. Atmos. Oceanic Technol., 12, 230–248.

Brown, R. A., L. R. Lemon, and D. W. Burgess,1978: Tornado detection by pulsed Dopplerradar. Mon. Wea. Rev., 106, 29–38.

Brown, R. A., and V. T. Wood, 1991: On theinterpretation of single-Doppler velocity patternswithin severe thunderstorms. Wea. Forecasting,6, 32–48.

Browning, K. A., and R. Wexler, 1968: The deter-mination of kinematic properties of a windfield using Doppler radar. J. Appl. Meteor.,7, 105–113.

Burgess, D. W., 1976: Single-Doppler radarvortex recognition. Part I. Mesocyclone signa-tures. Preprints, 17th Conf. on Radar Meteo-rology, (Seattle, USA, Amer. Meteor. Soc.),pp. 97–103.

Carbone, R. E., and F. D. Marks, Jr., 1988: Velocitytrack display (VTD): a real-time application forairborne Doppler radar data in hurricanes. Pre-prints, 18th Conf. on Hurricanes and TropicalMeteorology (San Diego, USA, Amer. Meteor.Soc.), pp. 11–12.

Caya, D., and I. Zawadzki, 1992: VAD analysis ofnonlinear wind fields. J. Atmos. Oceanic Technol.9, 575–587.

Rec

ent P

rogr

ess

in A

tmos

pher

ic S

cien

ces

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

NA

TIO

NA

L T

AIW

AN

UN

IVE

RSI

TY

on

01/1

9/15

. For

per

sona

l use

onl

y.

3 February 2009 9:2 B-582 ch16


Chan, J. C. L., and W. M. Gray, 1982: Tropicalcyclone movement and surrounding flow relation-ships. Mon. Wea. Rev., 110, 1354–1374.

Chen, Y., and M. K. Yau, 2001: Spiral bands in asimulated hurricane. Part I: Vortex Rossby waveverification. J. Atmos. Sci., 58, 2128–2145.

Davies-Jones, R. P., R. J. Trapp, and H. B. Blue-stein, 2001: Tornadoes and tornadic storms. InSevere Convective Storms. ed. C. A. Doswell(Amer. Meteor. Soc.), pp. 167–221.

Desrochers, P. R., and F. I. Harris, 1996: Interpre-tation of mesocyclone vorticity and divergencestructure from single-Doppler radar. J. Appl.Meteor., 35, 2191–2209.

Dodge, P., F. D. Marks, Jr., J. Gamache, J. Griffin,and N. Griffin, 1995: EVTD Doppler radar ana-lyses of the inner core of Hurricane Olivia (1994).Preprints, 27th Conf. on Radar Meteorology, (9–13 Oct. 1995, Vail, CO, USA, Amer. Meteor.Soc.), pp. 299–301.

Dodge, P., R. W. Burpee, and F. D. Marks Jr., 1999:The kinematic structure of a hurricane with sealevel pressure less than 900 mb. Mon. Wea. Rev.,127, 987–1004.

Donaldson, R. J., Jr., 1970: Vortex signature reco-gnition by a Doppler radar. J. Appl. Meteor., 9,661–670.

Donaldson, R. J., Jr., 1991: A proposed techniquefor diagnosis by radar of hurricane structure.J. Appl. Meteor., 30, 1636–1645.

Dong, K., and C. J. Neumann, 1986: The relati-onship between tropical cyclone motion and envi-ronmental geostrophic flows. Mon. Wea. Rev.,114, 115–122.

Dowell, D. C., C. R. Alexander, J. Wurman, andL. J. Wicker, 2005: Centrifuging of hydrometeorsand debris in tornadoes: radar-reflectivity pat-terns and wind-measurement errors. Mon. Wea.Rev., 133, 1501–1524.

Franklin, J. L., S. J. Lord, S. E. Feuer, and F. D.Marks, 1993: The kinematic structure of Hur-ricane Gloria (1985) determined from nested ana-lyses of dropwindsonde and Doppler radar data.Mon. Wea. Rev., 121, 2433–2451.

Gall, R., and J. Tuttle, 1999: A single radartechnique for estimating the wind in tropicalcyclones. Bull. Amer. Meteor. Soc., 80, 653–668.

Gamache, J. F., J. S. Griffin, P. Dodge, andN. F. Griffin, 2004: Automatic Doppler ana-lysis of three-dimensional wind fields in hurricaneeyewalls. Preprints, 26th Conf. on Hurricaneand Tropical Meteorology, 5D4, (3–7 May 2004,Miami, FL, USA, AMS).

George, J. E., and W. M. Gray, 1976: Tropicalcyclone motion and surrounding parameter rela-tionships. J. Appl. Meteor., 15, 1252–1264.

Gray, W. M., 2003: On the transverse circulationof the hurricane. In Cloud Systems, Hurricanes,and the Tropical Rainfall Measuring Mission(TRMM): A Tribute to Dr Joanne Simpson, eds.W.-K. Tao and R. Adler, pp. 149–164.

Griffin, J. S., R. W. Burpee, F. D. Marks, Jr., andJ. L. Franklin, 1992: Real-time airbone analysisof aircraft data supporting operational hurricaneforecasting. Wea. Forecasting, 7, 480–490.

Guinn, T. A., and W. H. Schubert, 1993: Hurricanespiral bands. J. Atmos. Sci., 50, 3380–3403.

Hanley, D., J. Molinari, and D. Keyser, 2001: A com-posite study of the interactions between tropicalcyclones and upper-tropospheric troughs. Mon.Wea. Rev., 129, 2570–2584.

Harasti, P. R., 2003: The hurricane volume velocityprocessing method. Preprints, 31st Conf. onRadar Meteorology, (Seattle, WA, USA, Amer.Meteor. Soc.), pp. 1008–1011.

Harasti, P. R., 2004: Separation of hurricane andenvironmental winds via single-Doppler radarvolumetric data analysis: results from HurricaneBret (1999). Preprints, 26th Conf. on Hurricanesand Tropical Meteorology, (Miami, FL, USA,Amer. Meteor. Soc.), pp. 124–125.

Harasti, P. R., and R. List, 1995: A single-Dopplerradar analysis method for intense axisymmetriccyclones. Preprints, 27th Conf. on Radar Meteo-rology, (Vail, CO, USA, Amer. Meteor. Soc.),pp. 221–223.

Harasti, P. R., and R. List. 2005: Principal com-ponent analysis of doppler radar data. Part I:Geometric connections between eigenvectors andthe core region of atmospheric vortices. J. Atmos.Sci., 62, 4027–4042.

Harasti, P. R., W.-C. Lee, and M. Bell, 2005:Rapid intensification of Hurricane Charley (2004)derived from WSR-88D data prior to landfall.Part II: HVVP adjustment for the environ-mental wind and central pressure retrieval. Pre-prints, 30th Int. Conf. on Radar Meteorology,(Albuquerque, New Mexico, USA, Amer. Meteor.Soc.), CD volume, paper J2J.3.

Harasti, P. R., W. C. Lee, and M. Bell, 2007:Real-time implementation of VORTRAC atthe National Hurricane Center. 33rd Conf. onRadar Meteorology, (6–11 August 2007, Cairns,Australia).

Harasti, P. R., C. J. McAdie, P. P. Dodge, W.-C.Lee, J. Tuttle, S. T. Murillo, and F. D. Marks,

Rec

ent P

rogr

ess

in A

tmos

pher

ic S

cien

ces

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

NA

TIO

NA

L T

AIW

AN

UN

IVE

RSI

TY

on

01/1

9/15

. For

per

sona

l use

onl

y.

3 February 2009 9:2 B-582 ch16


Jr., 2004: Real-time implementation of single-Doppler analysis methods for tropical cyclones:algorithm improvements and use with WSR-88Ddisplay data. Wea. Forecasting, 19, 219–239.

Holland, G. J., 1984: Tropical cyclone motion: acomparison of theory and observation. J. Atmos.Sci., 41, 68–75.

Houze, R. A. Jr., S. A. Rutledge, M. I. Biggerstaff,and B. F. Smull, 1989: Interpretation of Dopplerweather radar displays of midlatitude mesoscaleconvective systems. Bull. Amer. Meteor. Soc.,70, 608–619.

Houze, R. A., Jr., S. S. Chen, W.-C. Lee, R. F.Rogers, J. A. Moore, G. J. Stossmeister, J. L.Cetrone, W. Zhao, and M. M. Bell, 2006: TheHurricane Rainband and Intensity Change Expe-riment (RAINEX): Observations and modelingof Hurricanes Katrina, Ophelia, and Rita (2005).Submitted to Bull. Amer. Meteor. Soc.

Jorgensen, D. P., 1984a: Mesoscale and convective-scale characteristics of mature hurricanes. PartI: General observations by research aircraft.J. Atmos. Sci., 41, 1268–1286.

Jorgensen, D. P., 1984b: Mesoscale and convective-scale characteristics of mature hurricanes. PartII. Inner core structure of Hurricane Allen (1980).J. Atmos. Sci., 41, 1287–1311.

Jou, B. J.-D., and S.-M. Deng, 1997: Deter-mination of typhoon center and radius ofmaximum wind by using Doppler radar. 22ndConf. on Hurricanes and Tropical Meteo-rology, (May 1997, Fort Collins, CO, USA),pp. 104–105.

Jou, B. J.-D., B.-L. Chang, and W.-C. Lee, 1994:Analysis of typhoon circulation using ground-based Doppler radar (in Chinese, with Englishabstract). Atmos. Sci., 22, 163–187.

Jou, B. J.-D., S.-M. Deng, and B.-L. Chang, 1996:Determination of typhoon center and radiusof maximum wind by using Doppler radar (inChinese, with English abstract). Atmos. Sci., 24,1–23.

Jou, B. J.-D., J.-M. Chou, P.-L. Chang, and W.-C.Lee, 2001: Landfall typhoons observed by TaiwanDoppler radar network: a case study of typhoonBillis (2000). Int. Conf. on Mesoscale Meteo-rology and Typhoons in East Asia, (Sept. 2001,Taipei, Taiwan), pp. 171–178.

Jou, B. J.-D., W.-C. Lee, S.-P. Liu, and Y. C.Kao, 2008: Generalized VTD (GVTD) retrievalof atmospheric vortex kinematic structure. PartI: Formulation and error analysis. Mon. Wea.Rev. (in press).

Koscielny, A. J., R. J. Doviak, and R. Rabin, 1982:Statistical considerations in the estimation ofdivergence from single-Doppler radar and appli-cations to prestorm boundary layer observations.J. Appl. Meteor., 21, 197–210.

Kossin, J. P., W. H. Schubert, and M. T. Mont-gomery. 2000: Unstable interactions between ahurricane’s primary eyewall and a secondary ringof enhanced vorticity. J. Atmos. Sci., 57, 3893–3917.

Lamb, H., 1932: Hydrodynamics, 6th edn. (Dover),732 pp.

Lee, J. J., T. Samaras, and C. Young, 2004: Pressuremeasurements at the ground in an F-4 tornado.Preprints, 22nd Conf. on Severe Local Storms.(Amer. Meter. Soc.), CD volume, paper 15.3.

Lee, J. L., Y.-H. Kuo, and A. E. MacDonald, 2003:The vorticity method: extension to mesoscalevertical velocity and validation for tropicalstorms. Quart. J. Roy. Meteor. Soc., 129, 1029–1050.

Lee, J. L., W.-C. Lee, and A. E. MacDonald,2006: Estimating vertical velocity and flow fromDoppler radar observations of tropical cyclones.Quart. J. Roy. Meteor. Soc., 132, 125–145.

Lee, W.-C., and M. M. Bell, 2004: Kinematicstructure of Hurricane Isabel (2003) near landfallfrom the Morehead City WSR-88D. Preprints,26th Conf. on Hurricanes and Tropical Meteo-rology, (Miami, FL, USA, Amer. Meteor. Soc.),pp. 566–567.

Lee, W.-C., and M. M. Bell, 2007: Rapid inten-sification, eyewall contraction and breakdownof Hurricane Charley (2004) Near Landfall.Geophys. Res. Lett., 34, L02802, doi:10.1029/2006GL027889.

Lee, W.-C., and F. D. Marks, Jr., 2000: Tro-pical cyclone kinematic structure retrieved fromsingle Doppler radar observations. Part II: TheGBVTD-simplex center finding algorithm. Mon.Wea. Rev., 128, 1925–1936.

Lee, W.-C., and J. Wurman, 2005: Diagnosedthree-dirmensional axisymmetric structure of theMulhall tornado on 3 May 1999. J. Atmos. Sci.,62, 2373–2393.

Lee, W.-C., P. R. Harasti, and M. Bell, 2002:An improved algorithm to resolve circulationsof wavenumber two tropical cyclones. Preprints,25th Conf. on Hurricane and Tropical Meteo-rology, (29 Apr.-3 May 2002, San Diego, CA,USA, Amer. Meteor. Soc.), pp. 611–612.

Lee, W.-C., P. R. Harasti, and M. Bell, 2006b:VORTRAC: a utility to deduce central pressure

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and radius of maximum wind of landfalling tro-pical cyclones using WSR-88D data. Preprints,60th Interdepartmental Hurricane Conference(20–24 Mar. 2006, Mobile, Alabama, Office of theFederal Coordinator for Meteorological Servicesand Supporting Research).

Lee, W.-C., F. D. Marks, Jr., and R. E. Carbone,1994: Velocity track display (VTD): a techniqueto extract primary vortex circulation of a tropicalcyclone in real time using single airborne Dopplerradar data. J. Atmos. Oceanic Technol., 11, 337–356.

Lee, W.-C., J.-D. Jou, P.-L. Chang, and S.-M.Deng, 1999: Tropical cyclone kinematic structureretrieved from single Doppler radar observations.Part I: Interpretation of Doppler velocity pat-terns and the GBVTD technique. Mon. Wea.Rev., 127, 2419–2439.

Lee, W.-C., J.-D. Jou, P.-L. Chang, and F. D.Marks, 2000: Tropical cyclone kinematicstructure retrieved from single-Doppler radarobservations. Part III: Evolution and structureof Typhoon Alex (1987). Mon. Wea. Rev., 128,3892–4001.

Lee, W.-C., P. R. Harasti, M. Bell, B. J.-D. Jou,and M.-H. Chang, 2006a: Doppler velocity signa-tures of idealized elliptical vortices. Terr. Atmos.Oceanic Sci., 17, 429–446.

Lemon, L. R., D. W. Burgess, and R. A. Brown,1978: Tornadic storm sirflow and morphologyderived from single-Doppler radar measurement.Mon. Wea. Rev., 106, 48–61.

Liou, Y.-C., 1999a: A preliminary study of retrievinglow level vortex circulation using a Lagrangiancoordinate and single Doppler radar data, Terr.Atmos. Oceanic Sci., 10, 321–340.

Liou, Y.-C., 1999b: Single radar recovery of cross-beam wind components using a modified movingframe of reference technique. J. Atmos. OceanicTechnol., 16, 1003–1016.

Liou, Y.-C., T.-C. C. Wang, W.-C. Lee, andY.-J. Chang, 2006: The retrieval of asym-metric tropical cyclone structures using Dopplerradar observations and the method of extendedGBVTD. Mon. Wea. Rev., 134, 1140–1160.

Mallen, K. J., M. T. Montgomery, and B. Wang,2005: Re-examining tropical cyclone near-coreradial structure using aircraft observations:implications for vortex resiliency. J. Atmos. Sci.,62, 408–425.

Marks, F. D., 2003: State of the science: radarview of tropical cyclones. In Radar and Atmos-pheric Science: A Collection of Essays in Honor

of David Atlas, eds. R. Wakimoto and R. Sriva-stava (Amer. Meteor. Soc.), pp. 33–74.

Marks, F. D., and R. A. Houze, 1987: Inner corestructure of Hurricane Alicia from Doppler radarobservations. J. Atmos. Sci., 44, 1296–1317.

Marks, F. D., R. A. Houze, and J. F. Gamache, 1992:Dual aircraft investigation of the inner core ofHurricane Norbert. Part I: Kinematic structure.J. Atmos. Sci., 49, 919–942.

Matejka, T., and R. C. Srivastava. 1991: Animproved version of the extended velocity-azimuth display analysis of single-doppler radardata. J. Atmos. Oceanic Technol, 8, 453–466.

McAdie, C. J., P. R. Harasti, P. Dodge, W.-C. Lee,S. Murillo, and F. D. Marks, Jr., 2001: Real-timeimplementation of tropical cyclone-specific radardata processing algorithms. Preprints, 30th Int.Conf. on Radar Meteorology, (Munich, Germany,Amer. Meteor. Soc.), pp. 468–470.

Montgomery, M. T., and R. J. Kallenbach, 1997:A theory for vortex Rossby-waves and its appli-cation to spiral bands and intensity changes inhurricanes. Quart. J. Roy. Meteor. Soc., 123,435–465.

Murillo, S. T., W.-C. Lee, and F. D. Marks, 2000:Evaluating the GBVTD-tropical center findingsimplex algorithm. Preprints, 24th Conf. on Hur-ricane and Tropical Meteorology (29 May–2 June2000, Fort Lauderdale, FL, USA, Amer. Meteor.Soc.), pp. 312–313.

Murillo, S. T., W.-C. Lee, F. D. Marks, and P.Dodge, 2002: Examining structural changes andcirculation center of Hurricane Danny (1997)using a single-Doppler radar wind retrieval tech-nique. Preprints, 25th Conf. on Hurricane andTropical Meteorology (29 April–3 May 2002, SanDiego, CA, USA, Amer. Meteor. Soc.), pp. 485–486.

Neldar, J. A., and R. Mead, 1965: A simplexmethod for function minimization. Comp. J., 7,308–313.

Qiu, C.-J., and Q. Xu, 1992: A simple adjointmethod of the wind analysis for single-Dopplerdata. J. Atmos. Oceanic Technol., 9, 588–598.

Rankine, W. J. M., 1901: A Manual of AppliedMechanics. 16th edn. (Charles Griff), pp. 574–578.

Ray, P., 1990: Convective dynamics. IN Radar inMeteorology, ed. D. Atlas (Amer. Metror. Soc.),pp. 348–390.

Rinehart, R., 1979. Internal storm motions from asingle non-Doppler weather radar. (NCAR/TN-146+STR), 262 pp.

Rec

ent P

rogr

ess

in A

tmos

pher

ic S

cien

ces

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

NA

TIO

NA

L T

AIW

AN

UN

IVE

RSI

TY

on

01/1

9/15

. For

per

sona

l use

onl

y.

3 February 2009 9:2 B-582 ch16


Rinehart, R., and E. T. Garvey, 1978: Three-dimensional storm motion detection by conven-tional weather radar. Nature, 273, 287–289.

Rotunno, R., 1979: A study in tornado-like vortexdynamics. J. Atmos. Sci., 36, 140–155.

Rotunno, R., and K. A. Emanuel, 1987: An air-seainteraction theory for tropical cyclones. Part II:Evolutionary study using a nonhydrostatic axi-symmetric numerical model. J. Atmos. Sci., 44,542–561.

Roux, F., F. Chane-Ming, A. Lasserre-Bigorry, andO. Nuissier, 2004: Structure and evolution ofintense tropical cyclone Dina near La Reunionon 22 January 2002: GB-EVTD analysis of singleDoppler radar observations. J. Atmos. OceanicTechnol., 21, 1501–1518.

Roux, F., and F. D. Marks, Jr., 1996: Extendedvelocity track display (EVTD): an improved pro-cessing method for Doppler radar observationsof tropical cyclones. J. Atmos. Oceanic Technol.,13, 875–899.

Schubert, W. H., M. T. Montgomery, R. K., Taft,T. A. Guinn, S. R. Fulton, J. P. Kossin, and J. P.Edwards, 1999: Polygonal eyewalls, asymmetriceye contraction, and potential vorticity mixing inhurricanes. J. Atmos. Sci., 56, 1197–1223.

Shapiro, A., S. Ellis, and J. Shaw, 1995: Single-Doppler retrievals with Phoenix 11 data: clear airand microburst wind retrievals in the planetaryboundary layer. J. Atmos. Sci., 52, 1265–1536.

Stirling, J., and R. M. Wakimoto, 1989: Mesoscalevortices in the stratiform region of a decayingmidlatitude squall line. Mon. Wea. Rev., 117,452–458.

Symthe, G. R., and D. S. Zrnic, 1983: Correlationanalysis of Doppler radar data and retrieval ofthe horizontal wind. J. Climate Appi. Meteor.,22, 297–311.

Tanamachi, R. L., H. B. Bluestein, W.-C. Lee,M. Bell, and A. Pazmany, 2006: Ground-basedvelocity track display (GBVTD) analysis of aW-band Doppler radar data in a tornado nearStockton, Kansas, on 15 May 1999. Mon. Wea.Rev. (accepted with revision).

Terwey, W. D., and M. T. Montgomery, 2002:Wavenumber-2 and wavenumber-m vortex rossbywave instabilities in a generalized three-regionmodel. J. Atmos. Sci., 59, 2421–2427.

Tuttle, J. D., and G. B. Foote, 1990: Determi-nation of the boundary layer airflow from a singleDoppler radar. J. Atmos. Oceanic Technol., 7,218–232.

Waldteufel, P., and H. Corbin, 1979: On the analysisof single-Doppler radar data. J. Appl. Meteor.,18, 532–542.

Wang, Y., 2002a: Vortex Rossby waves in a numeri-cally simulated tropical cyclone. Part I: Overallstructure, potential vorticity and kinetic energybudgets. J. Atmos. Sci., 59, 1213–1238.

Wang, Y., 2002b: Vortex Rossby waves in a nume-rically simulated tropical cyclone. Part II: Therole in tropical cyclone structure and intensitychanges. J. Atmos. Sci., 59, 1239–1262.

Wang, Y., and C.-C. Wu, 2004: Current under-standing of tropical cyclone structure andintensity changes: a review. Meteorol. Atmos.Phys., 87, 257–278.

Ward, N. B., 1972: The exploration of certain fea-tures of tornado dynamics using a laboratorymodel. J. Atmos. Sci., 29, 1194–1204.

Willoughby, H. E., 1992: Linear motion of ashallow-water barotropic vortex as an initial-value problem. J. Atmos. Sci., 49, 2015–2031.

Willoughby, H. E., and M. B. Chelmow, 1982:Objective determination of hurricane tracks fromaircraft observations. Mon. Wea. Rev., 110,1298–1305.

Winn, W. P., S. J., Hunyady, and G. D. Aulich,1999: Pressure at the ground in a large tornado.J. Geophy. Res., 104: D18, 22067–22082.

Wood, V. T., 1994: A technique for detecting atropical cyclone center using a Doppler radar.J. Atmos. Oceanic Technol., 11, 1207–1216.

Wood, V. T., and F. D. Marks, Jr., 1989: Hur-ricane Gloria: simulated land-based Doppler velo-cities reconstructed from airborne Doppler radarmeasurements. Preprints, 18th Conf. on Hur-ricane and Tropical Meteorology (San Diego,USA, Amer. Meteor. Soc.), pp. 115–116.

Wood, V. T., and R. A. Brown, 1992: Effects ofradar proximity on single-Doppler velocity of axi-symmetric rotation and divergence. Mon. Wea.Rev., 120, 2798–2807.

Wurman, J., 2002: The multiple vortex structure ofa tornado. Wea. Forecasting, 17, 473–505.

Wurman, J., and S. Gill, 2000: Finescale radar ob-servations of the Dimmitt, Texas (2 June 1995),tornado. Mon. Wea. Rev., 128, 2135–2164.

Wurman, J., and T. Samaras, 2004: Comparison ofin-situ and DOW Doppler winds in a tornado andRHI vertical slices through 4 tornadoes during1996–2004. Preprints, 22nd Conf. on Severe LocalStorms (Amer. Meteor. Soc.), CD volume, paper15.4.

Rec

ent P

rogr

ess

in A

tmos

pher

ic S

cien

ces

Dow

nloa

ded

from

ww

w.w

orld

scie

ntif

ic.c

omby

NA

TIO

NA

L T

AIW

AN

UN

IVE

RSI

TY

on

01/1

9/15

. For

per

sona

l use

onl

y.

3 February 2009 9:2 B-582 ch16


Wurman, J., J. M. Straka, E. N. Rasmussen,M. Randall, and A. Zahrai, 1997: Design anddeployment of a portable, pencil-beam, pulsed,3-cm Doppler radar. J. Atmos. Oceanic Technol.,14, 1502–1512.

Xu, Q., C. Qiu, and J. Yu, 1994: Adjointmethod retrieving of horizontal wind from single-Doppler reflectivity measured during Phoenix II.J. Atmos. Oceanic Technol., 11, 275–288.

Xu, Q., and C. J. Qiu, 1994: Simple adjointmethods for single-Doppler wind analysis with astrong constraint of mass conservation. J. Atmos.Oceanic Tech., 11, 289–298.

Yamauchi, H., O. Suzuki, and Akaeda, 2002: Asym-metry in wind field of Typhoon 0115 analyzed

by triple Doppler radar observations. Preprints,Int. Conf. on Mesoscale Convective Systems andHeavy Rainfall/Snowfall in East Asia (Tokyo,Japan, Japan Science and Technology Corpo-ration and Chinese Academy of MeteorologicalSciences), pp. 245–250.

Zawadzki, I. I., 1973: Statistical properties ofprecipitation patterns. J. Appi. Meteor., 12,459–472.

Zhao, K., B. J.-D. Jou, Y.-J. Pan, and W.-Z.Ge, 2007: Structural characteristics of a meso-vortex in Taiwan Strait from single Dopplerradar analysis. Acta Meteor Sinica (in Chinese),in press.

Rec

ent P

rogr

ess

in A

tmos

pher

ic S

cien

ces

Dow

nloa

ded

from

ww

w.w

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omby

NA

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NA

L T

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IVE

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01/1

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