Tropical Cyclone Kinematic Structure Retrieved from Single-Doppler Radar Observations

OCTOBER 1999 LEE ET AL. 2419

Tropical Cyclone Kinematic Structure Retrieved from Single-Doppler Radar Observations. Part I: Interpretation of Doppler Velocity Patterns and the GBVTD Technique

WEN-CHAU LEE National Center for Atmospheric Research,* Boulder, Colorado

ϩ BEN JONG-DAO JOU,PAO-LIANG CHANG, AND SHIUNG-MING DENG# Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan

(Manuscript received 22 April 1998, in ®nal form 11 September 1998)

ABSTRACT Deducing the three-dimensional primary circulation of landfalling tropical cyclones (TCs) from single ground- based Doppler radar data remains a dif®cult task. The evolution and structure of landfalling TCs and their interactions with terrain are left uncharted due to the lack of dual-Doppler radar observations. Existing ground- based single-Doppler radar TC algorithms provide only qualitative information on axisymmetric TC center location and intensity. In order to improve understanding of the wind structures of landfalling TCs using the widely available WSR-88D data along the U.S. coastal region, a single ground-based radar TC wind retrieval technique, the ground-based Velocity Track Display (GBVTD) technique, is developed. Part I of this paper presents 1) single-Doppler velocity patterns of analytic, asymmetric TCs, 2) derivation of the GBVTD technique, and 3) evaluation of the GBVTD-retrieved winds using analytic TCs. The Doppler velocity patterns of asymmetric TCs display more complex structure than their axisymmetric counterparts. The asymmetric structure of TCs can be inferred qualitatively from the pattern (or curvature) of the zero Doppler velocity line and the position and shape of the Doppler velocity dipole. However, without knowing the axisymmetric portion of the TC circulation, it is extremely dif®cult to extract quantitative information from these similar Doppler velocity patterns. Systematic evaluations on the GBVTD-retrieved winds show good agreement compared with the original analytic wind ®elds for axisymmetric ¯ows plus mean wind and/or angular wavenumber 1, 2, and 3 asymmetry. It is also shown that the GBVTD technique retrieves wind maxima that are not directly observed (perpendicular to the radar beams) because the GBVTD technique uses the Doppler velocity gradient, not the observed maxima, to retrieve wind maxima. The success of the GBVTD-retrieved winds and understanding their characteristics provide the theoretical basis to nowcast TC kinematic structure.

1. Introduction man life and property than all other natural disasters (Anthes 1982). Recent examples including Hurricane Severe tropical cyclones1 (TCs) are one of the most Iniki (1992), Hurricane Andrew (1992), Typhoon Herb devastating natural disasters along coastal regions in the (1996), and Typhoon Paka (1997) produced massive tropics. Historically, severe TCs cause more loss of hu- damage on the island of Kauai, southern Florida, Tai- wan, and Guam, respectively. A TC has long been recognized as a large cyclonic 1 An intense tropical cyclone where maximum winds exceed 34 m rotating body of winds characterized by a calm region sϪ1 is known as hurricane, typhoon, or cyclone, depending on its near its circulation center. The intensity of a TC is usu- geographical region. ally classi®ed by its surface maximum wind and/or min- * The National Center for Atmospheric Research is sponsored by imum central pressure. Our understanding of TC cir- the National Science Foundation. ϩ Current af®liation: Wu-Fen-Shan Radar Station, Central Weather culations grew with the advance of atmospheric ob- Bureau, Taipei, Taiwan. serving systems over the years, from rawinsondes, air- # Current af®liation: System Engineering Division, Institute for craft in situ measurements, dropsondes, satellite, and Information Industry, Taipei, Taiwan. Doppler radars. Although the intensity of a TC can be inferred from one or more measurements from these Corresponding author address: Dr. Wen-Chau Lee, National Center instruments, the detailed three-dimensional kinematic for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. structure of a TC can only be revealed by Doppler ra- E-mail: [email protected] dars.

᭧ 1999 American Meteorological Society 2420 MONTHLY WEATHER REVIEW VOLUME 127

Doppler radar has become an essential tool in ob- hypothetical TC is essentially identical to those pro- serving mesoscale vortices since its ®rst meteorological duced by mesocyclones. use in the late 1950s (Metcalf and Glover 1990). Since Wood and Brown (1992, hereafter referred to as WB) the early 1980s, the tail Doppler radar on the National systematically studied the Doppler velocity pattern of Oceanic and Atmospheric Administration (NOAA) WP- axisymmetric TCs consisting of both the tangential and 3D has collected Doppler radar data during hurricane radial ¯ows based on a Rankine (1901) vortex. Quan- reconnaissance and research missions. Dual-Doppler ra- titatively, a Rankine combined vortex (e.g., Lemon et dar data can be collected from two perpendicular ¯ight al. 1978) is characterized by two ¯ow regimes, a solid- legs (e.g., a ``®gure 4'' pattern) equipped with a classical body rotation, with VT proportional to R (radius), sur- (steerable) antenna or a single leg equipped with a dual- rounded by a potential vortex, with VT proportional to beam (i.e., the ``French'') antenna. Then, the three-di- 1/R. The maximum wind, Vmax, is at the radius of max- mensional TC internal structure can be reconstructed imum wind (Rmax), the boundary between the regimes. (e.g., Marks and Houze 1984, 1987; Marks et al. 1992; Wood and Brown (1992) concluded the following Roux and Viltard 1995). Although dual-Doppler radar Doppler velocity characteristics for axisymmetric TCs. derived TC structures have high research value, their 1) Single-Doppler velocity patterns of an axisymmetric real-time application capability is limited due to the rotation and/or divergence vary as a function of an elaborate data processing requirements. aspect ratio, A robust airborne single-Doppler wind retrieval tech- core diameter nique, the velocity track display (VTD), was proposed ␣ ϭ . (1) to deduce the primary circulation of TCs in real time distance from the radar to the TC center from an airborne Doppler radar scanning in a track- 2) The Doppler velocity maxima rotate counterclock- orthogonal mode (Lee et al. 1994). The VTD technique wise (clockwise) as axisymmetric out¯ow (in¯ow) and its extension, extended VTD (EVTD; Roux and superimposed on an axisymmetric rotational ¯ow. Marks 1996), run in real time on board the NOAA WP- The magnitude of the axisymmetric rotation and ra- 3Ds' workstations (Grif®n et al. 1992). The VTD- and dial ¯ows can be estimated from the displacement EVTD-derived winds are transmitted to the NOAA of the extreme Doppler velocities. Tropical Prediction Center located in Miami, Florida, 3) The apparent TC center (de®ned as the midpoint of via satellite in near±real time, providing a critical piece the chord connecting two extreme positive and neg- of information to the forecasters. ative Doppler velocity maxima) approaches the true In comparison, ground-based Doppler radars did not TC center when ␣ → 0. observe many TCs until recently and the dual-Doppler 4) The true TC center always falls on a circle passing radar observations of TCs are still rare events because through the radar and two Doppler velocity extremes. of the large baseline between Doppler radars. Dual- The work by Baynton (1979) and WB provide the Doppler radar analyses are further restricted by the lim- basis of using pattern recognition to estimate the inten- iting domain that usually covers only a portion of a TC sity and center position of an axisymmetric TC, which (e.g., Jou et al. 1997). Hence, single-Doppler radar are valuable resources for forecasters in making deci- techniques remain the primary tool for analyzing land- sions in real time. Asymmetric TCs, however, have been falling TCs. Single ground-based Doppler radar TC commonly observed by reconnaissance aircraft since techniques have their roots in a more general mesocy- 1950s (e.g., LaSeur and Hawkins 1963; Hawkins and clone Doppler radar signature. Through a pattern rec- Rubsam 1968; Marks and Houze 1987; Marks et al. ognition approach, Donaldson (1970) found that a vor- 1992). Stewart and Lyons (1996) ®rst documented the tex produced a Doppler velocity dipole signature with asymmetric Doppler velocity patterns in Typhoon Ed opposite parity. This dipole Doppler velocity signature (1993) from a Weather Surveillance Radar-1988 Dopp- can be identi®ed by a Doppler radar scanning azimuth- ler (WSR-88D) in Guam. Nevertheless, the general ally in a plane position indicator (PPI) mode. The az- characteristics of Doppler velocity patterns in asym- imuthal shear of radial velocities was used to identify metric TCs remain unclear. Therefore, it is bene®cial to potentially intense atmospheric vortices on the radar expand the analytic Doppler velocity patterns for axi- scope, which led to the tornado vortex signature (Brown symmetric TCs in WB to include asymmetric TCs and et al. 1978). examine their properties. Although pattern recognition The ®rst application of the Doppler velocity vortex is an ef®cient and practical way in nowcasting TCs, it signature to tropical cyclones was proposed by Baynton usually uses only point values, such as zero and peak (1979). The Doppler velocity pattern of a mean Atlantic Doppler velocities, to estimate TC center and intensity hurricane (Shea and Gray 1973) was constructed by qualitatively. There is currently no ability to reconstruct assuming a hypothetical coastal S-band (10-cm wave- the horizontal TC primary circulations from single- length) Doppler radar with a 2Њ beamwidth located 200 Doppler PPI scans using a pattern recognition method. km away from the hurricane center. Baynton (1979) In developing the velocity±azimuth display (VAD), demonstrated that the Doppler velocity pattern of this Lhermitte and Atlas (1962) ®rst used all available Dopp- OCTOBER 1999 LEE ET AL. 2421 ler velocities in a PPI scan, in conjunction with a linear ¯ow model, to yield quantitative information about weather events with winds varying linearly in space (also Caton 1963; Browning and Wexler 1968; Matejka and Srivastava 1991). Using a similar concept, but assuming a circular wind model, the VTD and EVTD techniques reconstructed the three-dimensional primary circulation of a TC from single airborne Doppler radar data. The axisymmetric and asymmetric tangential wind structures of Hurricane Gloria (1985) and Hugo (1989) were retrieved and compared favorably with in situ measurements. Airborne Doppler radar observations are limited by the ¯ight duration and operational restrictions over land. Therefore, airborne Doppler radar is more suitable to observe TCs over the open ocean. The completion of the WSR-88D network along the U.S. coastal region provides continuous surveillance capability of TC ac- tivity up to ϳ400 km (low-level re¯ectivity factor only) and within ϳ150 km (re¯ectivity and Doppler velocity) of the coastline. The WSR-88Ds offer a powerful tool for monitoring landfalling TCs should a VTD-type technique be implemented. Owing to different scanning geometry between the airborne and ground-based Doppler radar, a more general form of the equations is required to accommodate a more complex geometrical relationship. This new form of equations is named the ground- based VTD (GBVTD) technique. Part I of this paper establishes the background and tools to retrieve TC circulation from single ground- FIG. 1. The GBVTD geometry and symbols. based Doppler radar data. We present the Doppler velocity patterns of analytic axisymmetric and asymmetric TCs, derive the GBVTD technique, and examine the closely follows Fig. A1 in WB with modi®cations on characteristics of the GBVTD-retrieved primary TC cir- several symbols and the use of a right-handed coordinate culation using analytic wind ®elds. Section 2 describes systems. Symbols and their de®nitions appearing in Fig. the underlying concepts and the formulation of the 1 are summarized in appendix A. GBVTD technique, while section 3 presents the construction of an analytic dataset. Section 4 discusses the b. Concept Doppler velocity signatures of analytic asymmetric TCs. The general characteristics, advantages, and limitations The GBVTD concept is illustrated in Fig. 2, which of the GBVTD-retrieved analytic TC circulation are takes advantage of the Doppler velocity signatures of shown in section 5. The aliasing of the asymmetric ra- atmospheric vortices (Donaldson 1970; Brown et al. dial ¯ow into tangential ¯ow is discussed in section 6 1978). The vortex circulation is assumed circular where and the conclusions in section 7. zero ␥ and ␺ are de®ned at A (Fig. 1). For an axisymmetric positive tangential wind (counterclockwise in the Northern Hemisphere), V , illustrated in Fig. 2a, the 2. The GBVTD technique T Doppler velocities on a GBVTD ring are zero at A (␺ a. Geometrical relationship ϭ 0) and C (␺ ϭ ␲) where the beam direction is perpendicular to the TC tangential winds. The Doppler ve- 2 A ground-based Doppler radar is ®xed in space in locities equal the tangential wind at B (␺ ϭ ␲/2) and contrast to an airborne Doppler radar, which moves in D(␺ ϭ 3␲/2), where the Doppler radar measures the space. Therefore, the GBVTD geometrical relationship full component of the tangential winds. Note that the illustrated in Fig. 1 is different from the VTD geomet- functional dependence of TC circulation is now with ␺, rical relationship shown in Lee et al. (1994). Figure 1 Ã not ␥. As a result, Vd/cos␾ (de®ned in the next section) reduced to ϪVT sin␺. For an axisymmetric negative radial wind (in¯ow), VR, illustrated in Fig. 2b, the Doppler 2 Hereafter, ``Doppler radar'' represents ``ground-based Doppler velocities on the GBVTD ring are equal to the radial radar'' throughout this paper unless stated otherwise. wind at A and C while zero Doppler velocities are ob- 2422 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 2. GBVTD concept for (a) axisymmetric rotation, (b) axisymmetric radial in¯ow, and (c) axisymmetric rotation and radial in¯ow. Left panels illustrate the circulation of axisymmetric TCs (gray arrows) and their corresponding Doppler velocities (black arrows). Right panels illustrate the distributions of Doppler velocities vs azimuth angle.

Ã served at B and D. So, Vd/cos␾ has the form of VR 2), the Doppler velocity extreme are located at B and cos␺. Combining these two axisymmetric ¯ow patterns D(␺ ϭ ␲/2 and 3␲/2, respectively), which do not yields a negative sine curve with a phase shift (Fig. 2c). coincide with F and G (␥ ϭ ␲/2 and 3␲/2, respec- The sign of this phase shift indicates in¯ow (positive tively) in Fig. 1. As a result, chord BD does not pass phase shift of maximum Vd) or out¯ow (negative phase through the TC center (except when sin␣max ϭ 0) that shift of maxima Vd) while the magnitude of the phase leads to distortions in the observed Doppler velocity shift is determined by VR/VT of the vortex on a particular patterns (Fig. 1 in WB). radius. The difference between the GBVTD and VTD tech-

As discussed in Lee et al. (1994), the tangential and niques lies in the relationship between ␥ and ␪ d (F. radial velocities in a real TC are usually asymmetric, Roux 1998, personal communication). In VTD, ␪ d re- composed of mean ¯ow and waves of all scales. As a mains at ␲/2 while ␥ varies from 0 to 2␲ on a constant Ã result, Vd/cos␾ will not hold the simple sine and cosine radius because radar scans are considered parallel to form on a GBVTD ring as in the axisymmetric ¯ow each other. In GBVTD, ␪ d and ␥ are dependent be- ®elds illustrated in Fig. 2 but will appear as a complex cause all beams in a PPI scan cannot be considered waveform that can be decomposed into Fourier com- parallel. This difference implies that GBVTD contains ponents. These Fourier components contain contribu- more information than VTD but the inherent uncer- tions from the individual wavenumbers of the tangential tainty is more complex (demonstrated in the next sec- and radial winds at each radius of a TC. tion). When sin␣max ഠ 0 (i.e., the vortex is far away Note that the ␺ and ␥ are identical only at A and from the radar or R K R T ), all beams in the PPI scan C, where sin␣max ϭ 0. Here, sin␣ is half of the aspect intersecting this ring can be treated as parallel beams ratio ␣ de®ned in WB. Due to the geometrical rela- and the GBVTD geometry reduces to the VTD ge- tionship in PPI scans as illustrated in WB (their Fig. ometry. Hence, B approaches F and D approaches G OCTOBER 1999 LEE ET AL. 2423

asymptotically. When R ഠ R T , B and D move toward of ␣max and ␺ (the derivation is provided in appendix C and coincide with C at the location of the radar (O) B for interested readers) as follows: when sin␣max ϭ 1. When sin␣max Ն 1, that is, the radar is either on or inside the GBVTD ring. As a result, 1 Ϫ cos␣max1 ϩ cos␣ max cos␣ ഠ cos2␺ ϩ (5) the full component of V T is not observed by any beams ΂΃΂΃22 but, instead, all Doppler velocities have signi®cant sin␣ ϭ sin␣max sin␺. (6) contributions from the radial wind, V R . When the ground-based radar is located at the vortex center Substituting (4), (5), and (6) into (3), we obtain (sin ), the geometrical relationship becomes ␣max ഠ ϱ Ã essentially the VAD technique (Browning and Wexler Vd 1968), which samples only the radial component of cos␾ a vortex. Therefore, in order to sample the full tangential component of the vortex circulation at radius 1 Ϫ cos␣max1 ϩ cos␣ max ഠ V cos(␪ Ϫ ␪ ) cos2␺ ϩ R suitable for GBVTD analysis, a ground-based MTM[ ΂΃22 Doppler radar has to be located outside this radius (i.e., R Ͼ R). T Ϫ sin(␪ Ϫ ␪ ) sin␣ sin␺ TM max ] c. Mathematical formulation Ϫ VTRsin␺ ϩ V cos␺. (7) Ã The Doppler velocity (Vd), computed from the tan- Since ␣max is a constant for a given radius, Vd/cos␾ is gential velocity (VT), radial velocity (VR), vertical ve- a function of ␺ only. Following the VTD derivation in Ã locity (w), terminal velocity (␷ t), and the mean ¯ow Lee et al. (1994), Vd/cos␾,VT, and VR can be written (VM) of a tropical cyclone, is expressed by in terms of a truncated Fourier series:

L Vd ϭ VM cos(␥ Ϫ ␪M) cos␾ Ϫ VT sin␺ cos␾ VÃ (␺)/cos␾ ϭ (A cosn␺ ϩ B sinn␺), (8) ϩ V cos␺ cos␾ ϩ (w Ϫ ␷ ) sin␾. (2) dnn͸ R t nϭ0 M The terminal velocity (␷ t) can be estimated from a VTTnTn(␺) ϭ (VC cosn␺ ϩ VS sinn␺), (9) Z±␷ t relationship. The vertical velocity w is unknown ͸ nϭ0 and is set to zero in this study. After removing the N contribution from w and ␷ t in (2), the remaining Doppler V (␺) ϭ (VC cosn␺ ϩ VS sin␺), (10) Ã RRnRn͸ velocity is de®ned as Vd while its horizontal projection nϭ0 is expressed by where VTCn (VRCn) and VTSn (VRSn) are the amplitude VÃ of the sine and cosine component of V (V ) for angular d ϭ V cos(␥ Ϫ ␪ ) Ϫ V sin␺ ϩ V cos␺. (3) T R cos␾ MMTR wavenumber n (hereafter angular wavenumber is referred to as wavenumber). In order to perform a least squares ®t, all terms on the Substituting (8), (9), and (10) to (7) and applying the right-hand side of (3) have to be expressed as a function following trigonometric identities, of ␺ only. The term (␥ Ϫ ␪ ) in (3) can be written as M 2 sinA cosB ϭ sin(A ϩ B) ϩ sin(A Ϫ B), ␪T ϩ ␣ Ϫ ␪M. As a result, 2 cosA cosB ϭ cos(A ϩ B) ϩ cos(A Ϫ B), VM cos(␥ Ϫ ␪M)

ϭ VM[cos(␪T Ϫ ␪M) cos␣ Ϫ sin(␪T Ϫ ␪M) sin␣]. (4) 2 sinA sinB ϭ cos(A Ϫ B) Ϫ cos(A ϩ B), Meanwhile, cos␣ and sin␣ can be expressed in terms we obtain the following equation:

VÃ L d ϭ (A cosn␺ ϩ B sinn␺) (11) ͸ nn cos␾ nϭ0

1 Ϫ cos␣ 1 ϩ cos␣ ϭ V cos(␪ Ϫ ␪ )max cos2␺ ϩϪ max sin(␪ Ϫ ␪ ) sin␣ sin␺ Ϫ VC sin␺ MTM[ ΂΃22TM max ] T 0 1 M ϩ VC cos␺ Ϫ {VC[sin(n ϩ 1)␺ ϩ sin(1 Ϫ n)␺] ϩ VS[cos(n Ϫ 1)␺ Ϫ cos(n ϩ 1)␺]} R 0 ͸ Tn Tn 2 nϭ1 1 N ϩ {VC[cos(n ϩ 1)␺ ϩ cos(n Ϫ 1)␺] ϩ VS[sin(n ϩ 1)␺ ϩ sin(n Ϫ 1)␺]}. (12) ͸ Rn Rn 2 nϭ1 2424 MONTHLY WEATHER REVIEW VOLUME 127

The approximation sign has been replaced by an VCR 013ϭ A ϩ A Ϫ VCR 2 (21) equal sign in (12) and in the rest of the equations in this paper. Matching corresponding coef®cients of VST 1204ϭ A Ϫ A ϩ A cosn␺ and sinn␺ in (12), we obtain ϩ (A024ϩ A ϩ A Ϫ VCR 1 Ϫ VCR 3) 1 A ϭ {V cos(␪ Ϫ ␪ )(1 ϩ cos␣ ) ϫ cos␣max ϩ VCR 3 (22) 0 2 MTM max

VCT 124ϭϪ2(B ϩ B ) ϩ VSR 1 ϩ VSR 3 (23) Ϫ VST 1 ϩ VCR 1} (13) VS 2A VC (24) 1 T 23ϭ Ϫ R 2 A1 ϭ (2VCR 0 Ϫ VST 2 ϩ VCR 2) (14) 2 VCT 23ϭϪ2B ϩ VSR 2 (25) 1 VS ϭ 2A Ϫ VC (26) B ϭ [VCϪ 2VCϪ 2V sin(␪ Ϫ ␪ ) T 34R 3 1 2 T 2 T 0 MTM VCT 34ϭϪ2B ϩ VSR 3. (27) ϫ sin␣max ϩ VSR 2] (15) The ignored terms (involving VR on the right-hand side) 1 are presented in (19)±(27) to illustrate the biases due to A ϭ {V cos(␪ Ϫ ␪ )(1 Ϫ cos␣ ) 2 2 MTM max the closure assumptions. The VM cos(␪T Ϫ ␪M) (hereafter referred as ``along- Ϫ VST 3 ϩ VST 1 ϩ VCR 3 ϩ VCR 1} (16) beam VM'') and VM sin(␪T Ϫ ␪M) (hereafter referred as 1 ``cross-beam VM'') are the component of VM parallel and A (n Ն 3) ϭ [VS Ϫ VS ϩ VC nTn2 Ϫ1 Tnϩ1 Rnϩ1 perpendicular to OT and there is not enough information to further separate VM and ␪M. The cross-beam VM in ϩ VCRnϪ1] (17) (20) is unknown and aliased into the mean tangential wind, V C . The error is the greatest when ␪ Ϫ ␪ ϭ 1 T O T M B (n Ն 2) ϭ [VC Ϫ VC ϩ VS Ϯ␲/2. Since VM sin␪M(z) is normally much smaller than nTn2 ϩ1 TnϪ1 Rnϩ1 the mean tangential wind VTC0, the uncertainty in VTC 0 ϩ VS ]. (18) should be small except in a fast moving, weak TC. RnϪ1 Substituting (19)±(25) into (9) and (10), we obtain the Equations (13)±(18) are the general expressions be- vortex tangential winds (up to wavenumber 3) and the tween the observed Doppler velocities and the tangential mean radial wind (wavenumber 0). and radial winds on a GBVTD ring. For n ϭ 0toL, there are 2L ϩ 1 equations but there are 4L unknowns,

VM, ␪M, VTC0, VRC0, VTCn, VTSn, VRCn, VRSn, for n ϭ d. Computational procedure 1toL Ϫ 1. Examination of the equations shows the cosine part of the tangential wind and the sine part of The computational procedure of the GBVTD tech- the radial wind of the same wavenumber have similar nique includes the following ®ve steps. form and cannot be separated without additional obser- 1) Interpolating from PPI to constant-altitude PPI vations. Therefore, this is an underdetermined problem (CAPPI). and not all the coef®cients can be uniquely determined 2) Identifying the TC center. from single-Doppler observations. In fact, there are many 3) Interpolating data from CAPPI to TC cylindrical co- ways to close this underdetermined set of equations. One ordinates (␺). possible closure assumption is to follow the discussions 4) Performing GBVTD analysis (on ␺ coordinates) on in Lee et al. (1994), who assumed that the asymmetric each radius and altitude to obtain GBVTD coef®- VR is much smaller than the corresponding VT. This is cients. equivalent to assigning weighting function for all VT's to 5) Tropical cyclone wind ®eld construction in ␥ coor- 1 and all asymmetric VR's to 0, which reduces the num- dinates from the GBVTD coef®cients. bers of unknowns from 4L to 2L ϩ 1. Only those terms in (13)±(18) that do not include asymmetric VR are re- A bilinear interpolation scheme (Mohr et al. 1986) is tained after the simple tropical cyclone model is applied. used in steps 1 and 3. There are many ways to de®ne When L is limited from 0 to 4 and the above simple TC centers, such as, re¯ectivity, minimum pressure, and TC model is applied, the unknowns are reduced to 9 circulations. In this study, the TC center is de®ned as and can be solved in terms of An and Bn as follows: the circulation center and its location is known. How to objectively identify a TC center in real situations is V cos( ) A A A VC VC (19) MTM␪ Ϫ ␪ ϭ 024ϩ ϩ Ϫ R 1 Ϫ R 3 beyond the scope of this paper and will be discussed in

VCT 013ϭϪB ϩ B Ϫ VMTMsin(␪ Ϫ ␪ ) Part II. When the GBVTD analysis is performed on CAPPI data at different altitudes, the three-dimensional ϫ sin␣max ϩ VSR 2 (20) structure of a TC can be constructed. OCTOBER 1999 LEE ET AL. 2425

3. Construction of the analytic dataset is not considered. The Doppler radar is located at the grid origin (0, 0), which is 60 km south of the TC center There are several ground-based Doppler radar data- (0, 60). This simpli®cation does not reduce the gener- sets of tropical cyclones available for evaluating the ality of the results because 1) the Doppler velocity pat- GBVTD technique. However, using real data brings in tern of an axisymmetric TC depends only on sin␣ , other factors, such as uncertainty in center location, bi- max where sin␣ between 0 and 1 is simulated at different ases by unknown vertical velocities and terminal ve- max R for a ®xed R , and 2) the Doppler velocity pattern of locities, and unevenly distributed targets, that are not T an asymmetric TC also depends on ␪ and ␪ where directly related to the performance of the GBVTD tech- T M both forms of the angle dependence can be simulated nique. Most important, the true wind ®eld is unknown by varying the phase angle of the asymmetric compo- in a real dataset. For testing, we use an analytic dataset nents and ␪ . instead of a real dataset because of the following rea- M sons. 1) The performance of the GBVTD technique can be evaluated against wind ®elds with any combination 4. Characteristics of Doppler velocity patterns of of VT, VR, and VM. 2) The effects of real data on GBVTD analytic tropical cyclones performance can either be isolated and/or simulated in The characteristics of the Doppler velocity patterns the analytic dataset. for a number of analytic TCs are discussed in this sec- The analytic dataset is based on a single-level, ide- tion. These idealized Doppler velocity patterns reveal alized Rankine combined vortex at sea level. The axi- many useful guidelines that can be used to interpret real symmetric V is constructed as T TCs. The discussions are focused on ®ve commonly R observed ¯ow regimes: 1) Rankine combined vortex, 2) VT ϭ Vmax, R Յ R max, (28) axisymmetric VT plus VM, 3) axisymmetric VT plus ΂΃Rmax wavenumber 1, 4) axisymmetric VT plus wavenumber 2, and 5) axisymmetric V plus wavenumber 3. The R T V ϭ V max , R Ͼ R , (29) asymmetric radial ¯ows are not included in the discus- T max΂΃R max sion since they have the same signatures as asymmetric where V and R are set to 50 m sϪ1 and 20 km, tangential ¯ows at a 90Њ phase difference. Their effect max max is discussed qualitatively in section 6. respectively. In real TCs, the VT pro®les usually drop off much slower outside Rmax than the prescribed Ran- kine vortex. The axisymmetric VR is constructed such a. Rankine combined vortex that out¯ow (in¯ow) is inside (outside) R while V max R The ¯ow ®elds, their amplitude, and Doppler velocity is zero at the TC center, Rmax, and in®nity. This pattern is similar to that documented in Jorgensen (1984a,b). signatures of the Rankine combined vortex are illus- Hence, trated in Fig. 3. The Doppler velocity patterns of the axisymmetric VT (Fig. 3e) con®rms those characteristics 1/2 VR ϭ C1max[(R Ϫ R)R], R Յ R max (30) shown in WB, including the opening of the Doppler velocity patterns on the far side of the vortex. Note that V ϭϪC (R Ϫ R )1/2R /R, R Ͼ R . (31) R 2 max max max the zero Doppler velocity line (thick solid line) is a

Here, C1 and C 2 are scale factors and are assigned to straight line that passes through the TC center at (0, 60). Ϫ1 0.5 Ϫ1 0.1 s and3m s , respectively, putting VR within The Doppler velocity pattern for axisymmetric VT in a reasonable range. The asymmetric TC circulations are Fig. 3c is the ``basic pattern'' that will be used to com- constructed by superimposing higher wavenumbers onto pare with Doppler velocity patterns of asymmetric TCs the basic axisymmetric circulation. The magnitude of in subsequent subsections. each wavenumber is set to 10 m sϪ1 where its phase The Doppler velocity pattern of a realistic axisym- varies from 0 to 2␲/n, where n is the wavenumber. Here, metric VR (Fig. 3d) is more complicated than the simple Ϫ1 VM has a ®xed amplitude of 10 m s and ␪M varies radial out¯ow pattern illustrated in WB. The thick zero from0to2␲. The resulting idealized TC contains the Doppler velocity circle centered at the TC center rep- simple Rankine combined vortex and combinations of resents the transition from in¯ow to out¯ow at the Rmax. the asymmetric VT, VR, and VM. Another zero Doppler velocity circle passing through Ordinarily, radar resolution in the azimuthal direction the radar and the TC center represents the zero projec- is degraded with increasing range from the radar owing tion of VR along each radar beam. to the widening of the radar beam. In this paper, how- Combining axisymmetric VT and VR (Figs. 3a and 3b), ever, the radar beam is considered to be in®nitesimally the resulting Doppler velocity pattern is shown in Fig. narrow such that radar resolution is perfect at all ranges. 4. The velocity dipole rotates clockwise (counterclock-

Additionally, the simulation model assumes that the ra- wise) outside (inside) the Rmax as shown in WB (their dar measurements are free of noise. This hypothetical Fig. 3) and illustrated in the GBVTD concept. The zero Doppler radar has a 120-km effective range and a high Doppler velocity line passes through the TC center that effective Doppler velocity range where velocity aliasing bulges clockwise (in the in¯ow region outside Rmax and 2426 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 3. (a) and (b) Axisymmetric VT and VR ¯ow ®elds, (c) and (d) their amplitudes, and (e) and (f) the corresponding Doppler velocity patterns. Positive (negative) Doppler velocities are contoured in solid (dashed) lines. The thick solid line represents the zero Doppler velocity contour. The hurricane and radar symbols indicate the true TC center and the radar location, respectively.

counterclockwise (in the out¯ow region inside Rmax. The b. Effect of VM turning point from clockwise to counterclockwise bulg- Ϫ1 ing is located at the Rmax where the in¯ow and out¯ow When a 10 m s northerly VM is superimposed on collide. This characteristic of the zero Doppler velocity the basic pattern (Fig. 3c), the ¯ow ®eld changes (Fig. line can be used in real time to identify the mean in¯ow/ 5a) but the overall Doppler velocity pattern (Fig. 5b) out¯ow pattern and the location of maximum axisym- remains the same (e.g., Brown and Wood 1991). The metric convergence/divergence on low-level PPI im- zero Doppler velocity line curves and shifts to the right ages. side (cf. with Fig. 3b) of the true TC center (0, 60) in- OCTOBER 1999 LEE ET AL. 2427

ler velocity dipole. The zero Doppler velocity line curves clockwise (counterclockwise) on the far side of the TC

center when the alongbeam VM is approaching (receding).

c. Effect of wavenumber 1 Figure 6 shows examples of the wavenumber 1 component located at azimuths 90Њ, 180Њ, and 270Њ superimposed on the basic ¯ow. The zero Doppler velocity line remains a straight line and apparent centers collocated with the true vortex center at (0, 60) as the phase

angle of wavenumber 1 peak wind rotates along Rmax. When wavenumber 1 is located at 90Њ and 270Њ (Figs. 6b and 6f), the Doppler velocity pattern is symmetric similar to the pattern of the basic pattern (Fig. 3b) while the difference is in the locations of the peak Doppler velocity dipole. A wavenumber 1 at 90Њ (270Њ) shifts the peak Doppler dipole north (south) of those in the basic pattern. When wavenumber 1 is located at 180Њ (Fig. 6d), the gradient toward the positive (negative) peak wind is

FIG. 4. The Doppler velocity pattern of axisymmetric VT plus VR. tightened (loosened). The Doppler velocity pattern of wavenumber 1 located at 0Њ is the mirror image of Fig. 6d (not shown). dicated by the hurricane symbol but passes through the The total wind pattern produced by a northerly mean apparent center.3 The straight Doppler velocity line goes wind (Fig. 5a) and by a wavenumber 1 located at 180Њ through the true vortex center having the magnitude of (Fig. 6c) are almost identical except for the locations the alongbeam VM. The wind ®eld and Doppler velocity of the apparent centers and the gradients of velocity in pattern of a southerly VM (not shown) are essentially the cross-beam direction. Although these subtle differ- mirror images of Figs. 5a and 5b across the x ϭ 0 line. ences are dif®cult to distinguish in the total wind ®elds, The total winds with easterly (westerly) VM are shown the differences in the corresponding Doppler velocity in Fig. 5c (Fig. 5e) where the asymmetric ¯ow ®elds patterns are apparent in the behavior of the zero line shifts 90Њ from their northerly (southerly) VM counter- and the cross-beam Doppler velocity gradients (Figs. 5b parts and the apparent center shifted southward (north- and 6d). It is important to compare those differences in ward) of the true circulation center at (0, 60). Note that Doppler velocity patterns in Figs. 5 and 6 that provide the zero line is a straight line through the true TC center critical information for identifying not only the circu- when VM is perpendicular to the radar viewing direction. lation center, but also the cause of the apparent wave- Contrary to intuition, the Doppler velocity patterns number 1 structural asymmetry. in Figs. 5d and 5f are not mirror images of each other.

Due to the fan beams in PPI scans, the cross-beam VM affects the Doppler velocities as a function of the beam d. Effect of wavenumber 2 angle. The primary difference resides in the peak mag- Figure 7 shows examples of the wavenumber 2 com- nitude of the Doppler velocity dipole. The westerly VM ponent superimposed on the basic pattern where two enhances (reduces) the basic ¯ow south (north) of the peak amplitudes are located at 90Њ (270Њ), 135Њ (315Њ), TC center while easterly VM does the reverse. When VM and 180Њ (0Њ). The total wind ®elds (left panels in Fig. is neither parallel nor perpendicular to the beam through 7) have an elliptical shape. The primary (minor) axis the TC center (not shown), the Doppler velocity patterns orients along the peaks (valleys) of the wavenumber 2 possess a mixture of characteristics described above de- component. The apparent circulation centers are col- pending on the alongbeam and cross-beam VM. located with the true center at (0, 60) (e.g., Figs. 7a, In summary, the effect of alongbeam VM essentially 7c, and 7e) while the zero Doppler velocity lines remain affects the characteristics of the zero Doppler velocity line straight through the true TC center. When the wave- (e.g., the location of the apparent center) while the cross- number 2 is located at 90Њ (270Њ), the southern wind beam VM primarily in¯uences the magnitude of the Dopp- maximum pulls the the dipole toward the radar, while the northern wind maximum elongates the dipole on the far side of the TC center (e.g., Fig. 7b). When the wave- 3 The apparent center is de®ned as the point where ground-relative number 2 is located at 0Њ (180Њ), the Doppler velocity velocity is zero. This de®nition is different from the apparent center pattern (Fig. 7f) is similar to the basic ¯ow (Fig. 3b) de®ned in WB. except that the dipole is horizontally elongated (Fig. 2428 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 5. The magnitude (left) and Doppler velocity pattern (right) of axisymmetric component of

VT plus VM where (a) and (b) northerly VM, (c) and (d) easterly VM, and (e) and (f) westerly VM.

7b). The chord connecting the dipole is not perpendic- and 170Њ (290Њ and 50Њ) are illustrated in Fig. 8. The ular to the zero Doppler velocity line for all wavenumber apparent centers are collocated at the true vortex center 2 phase angles (e.g., Fig. 7d) except for the two sym- (0, 60) where the straight zero Doppler velocity line metric patterns illustrated previously. This characteristic passes through. The symmetric Doppler velocity pat- distinguishes wavenumber 2 ¯ows from axisymmetric terns in Fig. 8b are similar to Fig. 7f but these patterns and wavenumber 1 ¯ows. are quite different between Figs. 8d and 7b. In general, the Doppler velocity patterns show an elongation on one side of the dipole (Fig. 8f) due to the asymmetry e. Effect of wavenumber 3 in wavenumber 3 ¯ows. The cross-beam gradient of the Examples of wavenumber 3 superimposed on the ba- Doppler velocities varies as the wavenumber 3 com- sic pattern at 90Њ (210Њ and 330Њ), 150Њ (270Њ and 30Њ), ponent advances in phase. OCTOBER 1999 LEE ET AL. 2429

FIG. 6. The magnitude (left) and Doppler velocity pattern (right) of axisymmetric VT plus

wavenumber 1 component of VT at (a) and (b) 90Њ, (c) and (d) 180Њ, and (e) and (f) 270Њ.

5. The GBVTD-retrieved winds on analytic TCs the GBVTD-retrieved circulations from these analytic TCs are presented to demonstrate that the GBVTD The Doppler velocity patterns presented in the pre- technique is robust in retrieving TC primary circu- vious section provides qualitative guidance to identify lations from the Doppler velocity patterns. The TC structure from a low-level PPI scan. Although GBVTD analysis is performed from R ϭ 1kmtoR each asymmetric component possesses distinct char- ϭ 60 km with radial increment of 1 km and the data acteristics in its Doppler velocity pattern, similar were interpolated onto 90 points on a constant radius Doppler velocity patterns resulted from different ring with azimuthal increment of 4Њ. An error analysis combinations of asymmetry ¯ow ®elds create dif®- is performed to document biases between the culties in distinguishing TC structures exclusively GBVTD-retrieved winds and their analytical coun- from these Doppler velocity patterns. In this section, terparts. 2430 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 7. The magnitude (left) and Doppler velocity pattern (right) of axisymmetric VT plus

wavenumber 2 component of VT at (a) and (b) 90Њ, (c) and (d) 135Њ, and (e) and (f) 180Њ.

a. Rankine combined vortex and VR are explicitly resolved in the governing equations and illustrated in the GBVTD concept (Fig. 2). The GBVTD-retrieved VT (Fig. 9a) and VR (Fig. 9b) from the Doppler velocities in Fig. 4 are nearly identical b. Basic pattern plus VM compared with the analytic VT and VR in Figs. 3a and

3c. The bias (in %) between the GBVTD-retrieved When VM is northerly (alongbeam direction), the winds (VT and VR) and the analytic winds are less than GBVTD-retrieved winds (Fig. 10a) reproduced their

2% in all sin␣max and azimuth angles around the TC corresponding analytic winds (Fig. 5a). Some distor-

(Figs. 9c and 9d). These near-perfect results are not tions exist at large sin␣max; for example, the straight 10 Ϫ1 surprising because the axisymmetric circulation of VT ms contour (at x ϭ 50 km) in the analytic winds (Fig. OCTOBER 1999 LEE ET AL. 2431

FIG. 8. The magnitude (left) and Doppler velocity pattern (right) of axisymmetric VT plus

wavenumber 3 component of VT at (a) and (b) 90Њ, (c) and (d) 150Њ, and (e) and (f) 170Њ.

5a) becomes a curved contour in the retrieved wind ®eld The systematic biases on the retrieved V T as a per- (Fig 10a). When VM has a cross-beam component (east- centage of the V M on different sin␣max and V M angles erly or westerly in this example), the biases in the re- are illustrated in Fig. 11a. Biases exist whenever a trieved total winds become apparent and the GBVTD cross-beam V M component exists (␪ M other than 0Њ retrieves only the basic pattern while the asymmetry and 180Њ) and the error increases toward their max- produced by VM is completely unrecoverable (compar- imum at ␪ M ϭ 90Њ and 270Њ. Also, as sin␣max increases, ing Figs. 10b and 10c with Figs. 5c and 5e). The dif- more of the cross-beam component is aliased into the ferences among the retrieved vortices in Figs. 10b,c and retrieved axisymmetric V T . The biases of the retrieved the basic vortex in Fig. 9c suggest that the cross-beam V M (Fig. 11b) are only a function of ␪ M . These char- component of VM aliased into the retrieved VT. acteristics can also be concluded from Eqs. (19) and 2432 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 9. The GBVTD-retrieved (a) axisymmetric component of VT, (b) axisymmetric component

of VR, (c) error in retrieved component of VT (%), and (d) error in retrieved component of VR (%). The thick solid line is 0 contour. Positive (negative) contours are solid (dashed) lines. The contour interval for (c) and (d) is 1%.

(20). These biases depend on the amplitude and di- trieves the wavenumber 1 structure via the observed rection of V M . Doppler velocity tendencies, not from the direct obser- vation of the peak wind. Although the characteristics of the wavenumber 1 c. Basic pattern plus wavenumber 1 structure is preserved in the GBVTD-retrieved wind The GBVTD-retrieved winds for the basic pattern ®eld, it is evident that some distortions exist. These plus wavenumber 1 VT at different phases are illustrated distortions are primarily due to performing least square in Fig. 12. Unlike the apparent wavenumber 1 asym- ®t on a nonlinear (␺) coordinate systems. In order to metry produced by VM, the true wavenumber 1 asym- assess the systematic biases under these circumstances, metry is retrieved by the GBVTD technique regardless the reconstructed wind ®elds were decomposed into the of the phase angle of wavenumber 1 (comparing Figs. azimuthal (␥) domain to compare with the analytic wind

12 and 6). Surprisingly, the GBVTD-retrieved wave- ®eld. The biases in retrieved axisymmetric VT as a func- number 1 structures at phase angle of 90Њ (Fig. 12a) and tion of sin␣max and the phase (␥) of wavenumber 1 are 270Њ (Fig. 12c) compared favorably with their analytic summarized in Fig. 13a. Only half of the domain from winds (Figs. 6a and 6e) where the Doppler velocity 90Њ to 270Њ phase angles are shown because of the sym- patterns show characteristics of axisymmetric vortices. metric nature of the biases.

These two cases are extremely challenging because the Axisymmetric VT is overestimated (underestimated) Doppler radar does not observe the peak amplitude di- when the peak wavenumber 1 is located to the north rectly (Figs. 6b and 6f). As a result, nowcasting TC (south) of the TC center and the worst error (20%) oc- properties based on the Doppler velocity pattern (Fig. curs when sin␣max ϭ 1. The GBVTD technique always 6b) will not only misplace the peak wind location but underestimates the wavenumber 1 amplitude. In contrast also underestimate its intensity. However, GBVTD re- to the biases in axisymmetric VT, the worst biases of OCTOBER 1999 LEE ET AL. 2433

the wavenumber 1 amplitude (Fig. 13b) occurs when the phase angles of the wavenumber 1 are 0Њ and 180Њ,

and the errors exceed 40% when sin␣max ϭ 1 (e.g., the GBVTD ring passes through radar). The wavenumber 1 phase biases shift toward south (north) on the south (north) of the center with peaks at phase angles of 45Њ, 135Њ, 225Њ, and 315Њ. Overall phase biases are small → with maximum bias around 12Њ when sin␣max 1. In summary, the wavenumber 1 amplitude bias is worse

than the biases in axisymmetric VT and phase of wavenumber 1. If a 20% bias in the retrieved wind is ac- ceptable, the GBVTD technique is able to reasonably

retrieve wavenumber 1 asymmetric TCs when sin␣max Ͻ 0.7.

d. Basic pattern plus wavenumber 2 The wavenumber 2 structures are well retrieved in all cases (cf. Figs. 14 and 7). The elliptical shape contours

are preserved quite well when sin␣max Ͻ 0.7. However, the distortion outside Rmax is apparent. The retrieved maximum wind closer to the radar is stronger and more elongated than the one away from the radar (e.g., Figs. 14a and 14b). Wavenumbers 1 and 3 of tangential winds have been generated in the retrieved winds due to aliasing where there was only a wavenumber 2 component in the analytical wind ®eld. The GBVTD technique pro- duces an arti®cial wavenumber 1 along the major axis of the ellipse (Fig. 14) where the peak (valley) is located closer to (away from) the radar. When comparing only the wavenumber 2 components, both the phase and amplitude of the retrieved component have errors compared with their analytic component. Nevertheless, retrieved wavenumbers 1 and 3 (from aliasing) compensate for most of the biases in retrieved wavenumber 2 to produce a plausible total wind ®eld.

e. Basic pattern plus wavenumber 3 The GBVTD-retrieved winds for axisymmetric and

wavenumber 3 VT at different phase angles are illustrated in Fig. 15. The wavenumber 3 structure is re-

covered well inside Rmax (e.g., sin␣max Ͻ 0.33). The peak amplitude of wavenumber 3 is severely reduced on the far side of the TC. However, the peak wind locations still compare well with their analytic counterparts (Figs.

8 and 15). Outside Rmax, only a wavenumber 2 pattern is apparent. As a result, only limited quantitative information of wavenumber 3 can be obtained from these

severely aliased results beyond sin␣max Ͼ 0.33.

6. Discussion When higher wavenumber tangential and radial ¯ows FIG. 10. The GBVTD-retrieved winds for axisymmetric com- are present in the idealized TC, systematic evaluation ponent of V T plus V M where (a) northerly V M , (b) easterly V M , and of errors in VT for each scenario becomes dif®cult. Ex- (c) westerly V M . amining (19)±(26) reveals that the higher wavenumber radial ¯ow aliases into both amplitude and phase of their 2434 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 11. Biases of the GBVTD-retrieved winds (%) to the magnitude of V M as a function of sin␣max and ␪ M : (a) axisymmetric V T ,

and (b) V M . tangential counterparts, as well as other wavenumbers gle-Doppler radar data. These basic Doppler velocity owing to the nonlinear interaction among waves of dif- patterns of asymmetric TCs form a useful database that ferent wavenumbers. For each wavenumber, the abso- enables forecasters to extract basic TC information from lute error in the GBVTD-retrieved VT increases as the the Doppler velocity data in real time. The results show ratio of VR/VT increases (not shown). There is no aliasing that asymmetric TC Doppler velocity patterns are sig- error when VR/VT ϭ 0. When both wavenumber 1 VT ni®cantly different from those of the axisymmetric TCs Ϫ1 and VR have equal amplitudes of 5 m s (e.g., Roux documented in WB. The zero Doppler velocity line and Marks 1996), the amplitude of the GBVTD-re- passes through the true TC center in all cases except Ϫ1 trieved VT wavenumber 1 varies from 0 to 10 m s when the alongbeam mean ¯ow (VM) exists and its shape while the phase varies between Ϯ90Њ, depending on the is an important indicator of general structure of TCs. In relative phases between wavenumber 1 VT and VR. Sim- addition, the zero Doppler velocity line becomes curved ilar results can be extended to higher wavenumbers. For only when there are axisymmetric VR and alongbeam equal amplitudes of VT and VR, the amplitude aliasing VM. The differences in signatures are subtle but can be characteristics is the same as in the wavenumber 1 case, distinguished by examining the characteristics of the but the phase aliasing reduces from Ϯ90Њ to Ϯ90/nЊ, zero Doppler velocity contour. because the pattern is repeated n times around a circle The apparent wavenumber 1 asymmetry due to VM for wavenumber n. When the amplitude of wavenumber and the true wavenumber 1 asymmetry produce similar

1 VR is greater than VT, the assumption of the GBVTD total wind patterns, but they can be distinguished by technique breaks down and the phase biases can be as examining the characteristics of the zero Doppler ve- large as Ϯ180Њ. locity contour and the cross-beam Doppler velocity gradients across the TC center. The Doppler velocity patterns for wavenumbers 2 and 3 are quite distinct from 7. Conclusions the basic pattern and wavenumber 1 patterns and can

Part I of this paper presents two important aspects of be easily identi®ed. The existence of asymmetric VT using ground-based single-Doppler radar data in TC re- only modi®es the cross-beam Doppler velocity gradient, search and nowcasting. 1) Asymmetric TCs have more but the zero Doppler velocity contour remains a straight complicated Doppler velocity patterns than axisym- line and passes through the true TC center. metric TCs and the qualitative storm structure can be The GBVTD technique provides an objective ap- identi®ed from these patterns. 2) Introducing the proach to estimate the primary circulation of TCs from GBVTD technique shows that the GBVTD technique is single-Doppler velocity patterns. The GBVTD tech- robust in estimating TC vortex structure from axisym- nique uses the least squares method to ®t the primary metric TC to wavenumber 3 asymmetry. TC circulation onto the Doppler velocities on a ring The Doppler velocity patterns summarized in this pa- with constant radius from the TC center. The along- per expand the scope to qualitatively identify TC struc- beam VM, axisymmetric VR, and VT can be retrieved for tures from axisymmetric to asymmetric TCs using sin- each radius and altitude. The asymmetric radial ¯ows OCTOBER 1999 LEE ET AL. 2435

FIG. 13. Biases of the GBVTD-retrieved winds (%) to their analytic

values as a function of sin␣max and ␪M: (a) axisymmetric VT, and (b) magnitude of wavenumber 1, and (c) phase of wavenumber 1.

FIG. 12. The GBVTD-retrieved winds for axisymmetric VT plus wave-

number 1 component of VT at (a) 90Њ, (b) 180Њ, and (c) 270Њ. 2436 MONTHLY WEATHER REVIEW VOLUME 127

FIG. 14. The GBVTD-retrieved winds for axisymmetric VT plus wave- FIG. 15. The GBVTD-retrieved winds for axisymmetric VT plus wave-

number 2 component of VT at (a) 90Њ, (b) 135Њ, and (c) 180Њ. number 3 component of VT at (a) 90Њ, (b) 150Њ, and (c) 170Њ. OCTOBER 1999 LEE ET AL. 2437

and the cross-beam VM are not resolved, but alias into coordinate system centered at the radar. These coordi- the tangential winds and along-beam VM. These quan- nate systems and corresponding wind components are titative TC structures cannot be retrieved from a pattern de®ned as follows. recognition method. O: the location of the ground-based Doppler radar The GBVTD technique retrieves good total wind in T: the center of the TC nearly all cases tested in this study. It has been shown R: the radial distance from the TC center to the that the GBVTD technique is able to retrieve unsampled ring at a constant altitude where the GBVTD wind maxima that were not directly observed by a analysis is performed Doppler radar because it retrieves wind maximum using E: the intersection of a radar beam and a constant the tendency embedded in the Doppler velocity pattern, radius ring not relying on the observed absolute maxima or minima. A, C: intersections of the radar beam (OT) and a ring This characteristic, in conjunction with the ability to of radius R identify basic TC structures, are the two most important B, D: intersections of radar beams tangent to the ring advantages of using the GBVTD technique over the of radius R where OB ⊥ TB and OD ⊥ TD conventional pattern recognition methods in nowcasting F, G: FG passes through T and is perpendicular to landfalling TCs. OT The quality of the GBVTD-retrieved wind ®elds RD: the radial distance between the radar and a strongly depends on sin␣max (the aspect ratio, R/RT), pulse volume which essentially determines the distortion of the RT: the radial distance between the radar and the GBVTD geometry owing to the proximity to the radar. storm center The upper bond of sin␣max for good wind retrieval is ␪d: the mathematical angle of the radar beam mea- further limited by the existence of asymmetric circu- sured counterclockwise from the east lations. For a 20% bias, the upper limits of sin␣max are z: vertical axis of the TC cylindrical coordinate, 0.7, 0.5, and 0.33 for wavenumbers 1, 2 and 3, respec- z ϭ 0 is sea level tively. These results form theoretical bases on how to ␾: the elevation angle of the radar beam interpret and apply the GBVTD results in future TC VT: the tangential velocity of the TC, positive coun- studies. terclockwise (clockwise) in the Northern In Part II of this series (Lee and Marks, manuscript (Southern) Hemisphere submitted to Mon. Wea. Rev.), an objective vortex center VR: the radial velocity of the TC, positive outward ®nding algorithm, the GBVTD-simplex algorithm, is from the TC center presented. Applications of the GBVTD to real land- Vd: Doppler velocity from the ground-based Dopp- falling TCs are the focus in Part III of this series (Lee ler radar, receding (approaching) velocities et al. 1999, manuscript submitted to Mon. Wea. Rev.), from the radar are positive (negative) where the three-dimensional structure of Typhoon Alex ␷ t: terminal fall speed of the precipitation parti- (1987) is retrieved from the GBVTD technique using cles, positive is downward the circulation center deduced from the GBVTD-sim- w: vertical velocity, positive is upward plex algorithm. VM(z): the magnitude of the mean wind ¯ow, which is a function of altitude Acknowledgments. The authors are grateful to Drs. ␪M(z): the direction of the mean wind ¯ow, which is Peter Hildebrand, Frank Marks, Frank Roux, Tammy a function of altitude Weckwerth; Mr. Peter Dodge; and two anonymous ␣: the angle subtended by OE and OT (∠TOE) reviewers for their valuable comments that greatly ∠ ␣max: the maximum ␣ at a given radius ( TOB) improved this paper. Ms. Susan Stringer helped in ␺: ∠OET; when ␺ ϭ 0 (A) and ␲ (C), the radar preparing ®gures and Ms. Jenifer Delaurant proofread beam is parallel to radius TE; when ␺ ϭ ␲/2 the manuscript. This research is partially supported (B) and 3␲/2 (D), the radar beam is normal to by the National Science Foundation of the United radius TE States and the National Science Council of Taiwan, ␥: the mathematical angle for E in the TC cylin- Republic of China, under Grant NSC 88-2111-M-002- drical coordinate 007-AP6. ␪T: the mathematical angle for TC center viewing from the radar APPENDIX A Z: radar re¯ectivity factor Coordinate Systems of the GBVTD Technique APPENDIX B The coordinate systems and symbols of the GBVTD technique are illustrated in Fig. 1 elsewhere in the text. The Approximation of cos␣ There are two coordinate systems: a cylindrical coor- From the triangle OTE in Fig. 1, sin␣ and cos␣ can dinate system centered at the TC and a radar spherical be written as follows: 2438 MONTHLY WEATHER REVIEW VOLUME 127

sin␣ sin␺ ϭ (B1) RRT R sin␣ ϭ sin␺ (B2) RT

sin␣ ϭ sin␣max sin␺ (B3)

21/2 cos␣ ϭ [1 Ϫ (sin␣max sin␺) ] . (B4) Since cos␣ cannot be explicitly expressed in terms of sinn␺ and cosn␺, an approximation form of cos␣ is constructed as follows. Table B1 lists the corresponding cos␣ with respect to ␺. It is clear that cos␣ goes two cycles while ␺ varies from 0 to 2␲ and can be approx- imated as a function of cos2␺. In addition, cos␣ is bounded between cos␣max and 1. Therefore, cos␣ has an amplitude of (1 Ϫ cos␣max)/2: 1 Ϫ cos␣ 1 ϩ cos␣ cos␣ ഠ maxcos2␺ ϩ max . (B5) 22 Figure B1 illustrates the approximation of cos␣ in (B5) and its analytical expression in (5) at three different

␣max:30Њ,60Њ, and 80Њ. The differences vary throughout the 0±2␲ interval for a given ␣max. It clearly indicates that the extreme errors of this approximation increase with ascending ␣max as illustrated in Table B2. The errors are less than 1% when ␣max is less than 45Њ. The errors exceed 10% when ␣max is beyond 75Њ. Therefore, the approximation becomes questionable only when ␣max is greater than 75Њ.

TABLE B1. The relationship among ␺, ␣, and cos␣. ␺␣cos␣ 0 0 1

␲/2 ␣max cos␣max ␲ 0 1

3␲/2 Ϫ␣max cos␣max 2␲ 0 1

TABLE B2. The maximum and minimum differences between cos␣ and its approximation at different ␣max.10-t202 ␣ (Њ) Max dif Min dif max FIG. B1. The distribution of cos␣ and its approximations (B5) as a 5 5.96046 ϫ 10Ϫ8 Ϫ1.78814 ϫ 10Ϫ6 function of ␺. 10 0.0 Ϫ2.90871 ϫ 10Ϫ5 15 0.0 Ϫ1.47581 ϫ 10Ϫ4 20 0.0 Ϫ4.68612 ϫ 10Ϫ4 Ϫ3 25 0.0 Ϫ1.15108 ϫ 10 REFERENCES 30 0.0 Ϫ2.40469 ϫ 10Ϫ3 35 0.0 Ϫ4.49365 ϫ 10Ϫ3 Anthes, R. A., 1982: Tropical CyclonesÐTheir Evolution, Structure 40 0.0 Ϫ7.74080 ϫ 10Ϫ3 and Effects. Meteor. Monogr., No. 41, Amer. Meteor. Soc., 208 45 0.0 Ϫ1.25309 ϫ 10Ϫ2 pp. 50 0.0 Ϫ1.94002 ϫ 10Ϫ2 Baynton, H. W., 1979: The case for Doppler radars along our hur- 55 0.0 Ϫ2.88888 ϫ 10Ϫ2 ricane affected coasts. Bull. Amer. Meteor. Soc., 60, 1014±1023. 60 0.0 Ϫ4.16355 ϫ 10Ϫ2 Brown, R. A., and V. T. Wood, 1991: On the interpretation of single- 65 0.0 Ϫ5.85824 ϫ 10Ϫ2 Doppler velocity patterns within severe thunderstorms. Wea. 70 0.0 Ϫ8.05405 ϫ 10Ϫ2 Forecasting, 6, 32±48. 75 0.0 Ϫ0.109077 , L. R. Lemon, and D. W. Burgess, 1978: Tornado detection by 80 0.0 Ϫ0.145381 pulsed Doppler radar. Mon. Wea. Rev., 106, 29±38. 85 0.0 Ϫ0.191243 Browning, K. A., and R. Wexler, 1968: The determination of kine- OCTOBER 1999 LEE ET AL. 2439

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