## Abstract

This paper is the third of a series that focuses on the applications of the ground-based velocity track display (GBVTD) technique and the GBVTD-simplex center finding algorithm developed in the previous two papers to a real tropical cyclone (TC). The evolution and structure of Typhoon Alex (1987), including full tangential winds, mean radial winds, one component of the mean flow, and their derived axisymmetric angular momentum and perturbation pressure fields are reconstructed from 16 volume scans (6.5 h of data with a 2-h gap) from the Civil Aeronautic Administration (CAA) Doppler radar while Typhoon Alex moved across the mountainous area in northern Taiwan.

This analysis retrieves a plausible and physically consistent three-dimensional primary circulation of a landfalling TC using a single ground-based Doppler radar. Highly asymmetric wind structures were resolved by the GBVTD technique where the maximum relative tangential wind at *z* = 2 km evolved from 52 m s^{−1} (before landfall), to less than 40 m s^{−1} (after landfall), to less than 35 m s^{−1} (entering the East China Sea). Alex’s eye began to fill with precipitation while its intensity decreased rapidly after landfall, a characteristic of circulations disrupted by terrain. The mean radial wind field revealed a layer of low-level inflow in agreement with past TC observations. The outward slope of the eyewall reflectivity maximum was consistent with the constant angular momentum contours within the eyewall. After Alex entered the East China Sea, its circulation became more axisymmetric.

The axisymmetric perturbation pressure field was retrieved using the gradient wind approximation, which, when used in conjunction with one or more surface pressure measurements within the analysis domain, can estimate the central pressure. The retrieved perturbation pressure fields at two time periods were compared with surface pressures reported in northern Taiwan. Considering the assumptions involved and the influence of terrain, good agreement (only 1–2-mb deviation) was found between them. This agreement indicates the relative quality of the GBVTD-retrieved axisymmetric circulation and suggests GBVTD-retrieved quantities can be useful in operational and research applications.

## 1. Introduction

Hurricane forecasts near landfall has been identified as one of the three research focuses in the U.S. Weather Research Program by the Fifth Prospectus Development Team (PDT-5) (Marks et al. 1998). In order to improve the understanding of the physical processes and provide the initial conditions for realistic numerical predictions, PDT-5 recognized the need to collect comprehensive three-dimensional (3D) datasets over the storm scale from all available observing platforms (e.g., Doppler radar, satellite, aircraft, etc.). Several of the research objectives identified by PDT-5 require detail descriptions of the tropical cyclone (TC) circulation near landfall. The recently completed coastal Weather Surveillance Radar-1988 Doppler (WSR-88D) network in the United States provides a unique opportunity to continuously monitor the precipitation intensity (via radar reflectivity factor) and Doppler velocity of landfalling TCs.

Although the 3D kinematic structures of TCs have been deduced from single- and/or dual-airborne Doppler radar analysis (e.g., Marks and Houze 1987; Marks et al. 1992; Lee et al. 1994; Roux and Viltard 1995; Roux and Marks 1996), past ground-based dual-Doppler radar analyses on a few TCs revealed primarily TC rainband structures, not the storm-scale circulation (e.g., Bluestein and Hazen 1989; Ishihara et al. 1986; Jou et al. 1997, 1999; Wang and Tseng 1999). This disparity results from the fact that the dual-Doppler analysis domain is usually limited only to a portion of the TC. Single-Doppler radar analysis techniques (e.g., Wood and Brown 1992) can estimate intensity and location of an axisymmetric TC from the Doppler velocity dipole signature. These Doppler velocity signatures (e.g., zero Doppler velocity line, Doppler velocity maxima, and the Doppler velocity gradient across the zero Doppler velocity line) are valuable information to infer TC circulations in operational forecast. However, Lee et al. (1999) illustrated that structures of asymmetric TCs may not be properly inferred via these basic Doppler velocity signatures due to the close proximity in these velocity patterns. This series of papers [including Lee et al. (1999, hereafter Part I) and Lee and Marks (2000, hereafter Part II)] attempts to bridge this gap by proposing and demonstrating a TC wind retrieval technique, the ground-based velocity track display (GBVTD) presented in Part I, that can be used to derive a plausible and consistent TC primary circulation (i.e., tangential winds) from the velocity data collected by a single ground-based Doppler radar.

The mathematical formulation of the GBVTD technique and its intrinsic assumptions and limitations were presented in Part I. Assuming a circular wind model and ignoring asymmetric radial winds,^{1} the GBVTD technique retrieves the axisymmetric tangential wind and radial wind, wavenumbers 1–3 of tangential wind, and the mean cross-TC flow, via a least squares fit of the observed radial velocities on each radius and altitude on a cylindrical coordinate centered at the TC. Part I showed that the GBVTD technique retrieved robust two-dimensional asymmetric TC primary circulations from the simulated single-Doppler radar observations of several analytic TCs, especially for the three lowest wavenumbers.

It is noted in Part I and other studies (e.g., Lee et al. 1994; Roux and Marks 1996) that accurately knowing the TC center critically determines the quality of the GBVTD-retrieved TC circulation. Here, the TC center is defined as the circulation center or the vorticity center of a TC (consistent with definition used in Parts I and II). It is demonstrated in Part II that a nominal 1–2-km error in the estimated TC center is required to keep the error in the GBVTD-derived asymmetric circulation below 20%. In order to achieve this goal, the GBVTD-simplex TC center finding algorithm was developed to objectively identify the TC center^{2} from single-Doppler radar data. When tested on analytical TCs with a known circulation center, Part II showed that the mean errors of the GBVTD-simplex determined TC centers were less than 500 m computed on a 1-km horizontal grid resolution. When the GBVTD-simplex algorithm was applied to Typhoon Alex (1987), the estimated uncertainty of the objectively determined center^{3} were between 1 and 2 km while the uncertainty can be as high as 3–4 km in some conditions discussed in Part II.

The purpose of this paper is twofold: 1) demonstrate that the GBVTD technique produced reasonable TC structures on a real TC, and 2) document the evolution and structure of Typhoon Alex under the influence of topography. Emphasis is placed on the structure and intensity change before, during, and after Alex’s landfall on northern Taiwan where terrain played a major role in modifying the storm structure. Although there were no aircraft in situ measurements to verify the retrieved kinematic quantities, the retrieved axisymmetric perturbation pressure is compared with the surface pressure reported at five stations in northern Taiwan to test the consistency of the wind retrieval.

Section 2 presents a synopsis of Typhoon Alex. Section 3 discusses the GBVTD analysis procedures and derived quantities. The evolution and asymmetric structure of Alex are presented in section 4. Section 5 illustrates the axisymmetric structure of Alex using the GBVTD-retrieved axisymmetric winds, angular momentum, and perturbation pressure. The perturbation pressures at two time periods are compared with surface pressure reports in section 6. Summary and conclusions are presented in section 7. Practical limitations of applying GBVTD technique to real data are discussed in the appendix.

## 2. Typhoon Alex

Typhoon Alex formed in the western Pacific, east of the Philippines on 23 July 1987. It moved northwest steadily along the southwestern edge of the subtropical high. Alex then moved NNW along the east coast of Taiwan. Alex’s intensity (maximum wind speed and central pressure) and storm speed from 1400 LST (LST = UTC + 8; hereafter, all times are LST) 24 July to 0800 28 July 1987 estimated from satellite images are illustrated in Fig. 1 [adapted from Wang (1987)]. Alex reached hurricane strength (maximum wind exceeding 34 m s^{−1}) about 1400 25 July 1987, and weakened to a tropical storm about 1100 27 July 1987. During this 3-day period, it moved generally NNW between 5 and 7 m s^{−1}.

The Civil Aeronautic Administration (CAA) Doppler radar is a C-band operational radar located at Chiang-Kai-Shek International Airport in northwestern Taiwan (marked by the radar symbol in Fig. 2). The characteristics of the CAA Doppler radar are summarized in Table 6 of Part II. The CAA Doppler radar has a maximum range of 120 km in the Doppler mode and completes a 10 elevation angle volume scan every 15 min. Alex’s center entered CAA’s Doppler range around 0400 27 July 1987 and left CAA’s Doppler range about 1100 27 July 1987. There were no observations between 0700 and 0917 due to a power outage at the CAA radar site. A total of 16 volume scans (0432–0647 and 0917–1032) were available for analysis.

The topography in northern Taiwan within 120 km of the CAA radar is quite variable (Fig. 2). The altitude of the central mountain range decreases from ≈3000 m south of CAA to ≈500 m east of CAA. As a result, circulations below 1-km altitude were partially blocked by the central mountain range east to southeast of the CAA radar. CAA Doppler radar data were edited to remove noise and ground contaminated data. The dual-pulse repetition frequency capability of the CAA radar extended the maximum unambiguous velocity to ±48 m s^{−1}. Hence, the velocity was easily dealiased because the Doppler velocities are between ±55 m s^{−1}.

Alex’s circulation center in each radar volume was objectively determined using the GBVTD-simplex TC center finding algorithm (Part II) and was computed in a 1-km grid in all three directions from 1 to *z* = 10 km altitude. However, in this case the center computation is unusable above 4 km because of missing data at higher altitudes. Please refer to Part II for computational details and limitations of the GBVTD-simplex algorithm.

There were 1–2-km variations in Alex’s computed centers from 1- to 3-km altitude. Our results suggest that radius of maximum wind (RMW) may vary with height, which is supported by the vertical tilt of the axisymmetric tangential winds and reflectivity factors with altitude. No subjective adjustments to the GBVTD-simplex derived TC centers were made in this study. We chose to use the GBVTD-simplex determined TC centers at 2-km altitude (above the terrain in northern Taiwan) (Table 1 and Fig. 2) and realized there were potential errors in the retrieved asymmetric components resulting from this assumption. Unfortunately, we cannot assess the impact of using a fixed TC center without independent information such as concurrent aircraft in situ data.

## 3. The GBVTD analysis and products

### a. The GBVTD analysis

The GBVTD analysis provides one component of the mean flow, axisymmetric tangential and radial winds, and the asymmetric tangential winds from single-Doppler radar data, as outlined in Part I. The Doppler radar data collected in a constant elevation angle mode are interpolated into constant-altitude plan position indicators (CAPPIs) in Cartesian coordinates with a 1-km grid spacing on all three axes. Instead of linearly interpolating data onto an evenly spaced azimuthal grid at a constant radius from the TC center as illustrated in Part I, we allowed all data points that fall into a 4-km-wide annulus to be included in the least squares fit. This modification not only ties results on adjacent radii together, reducing the variations on adjacent radii seen in Lee et al. (1994), but also reduces the influence of outliers in the least squares fit in data-sparse regions. In this study, we allowed up to wavenumber 3 asymmetry in the GBVTD-retrieved tangential winds (i.e., a maximum of nine coefficients in the least squares fit). The actual truncation of the Fourier series on each radius depends on the largest single data gap (Table 2) on a given radius and the geometric factor (sin*α*_{max}, defined as the ratio of radius to the TC center, *R,* and the distance between radar and TC, *R*_{T}) described in Part I. These criteria reduce the potentially incoherent behavior of higher wavenumbers (especially wavenumbers 2 and 3) over large data gaps (inherent in real data) and ensure that the errors due to the geometric distortion are smaller than 20%. However, discontinuities of the retrieved wind fields occur when the truncation of the Fourier series differs on adjacent radii.

The GBVTD analysis is computed on a distorted coordinate system *ψ,* while its relationship with the mathematic coordinates, *θ,* is derived in Part I. Deviation between *ψ* and *θ* increases as sin*α*_{max} increases (i.e., as *R* increases for a fixed *R*_{T}). Examples of the curve fit in the distorted GBVTD coordinates, *ψ* (left column), and the mathematic coordinates, *θ* (right column), and the decomposed Fourier series (middle column) at radii 20, 35, and 50 km and altitude 2 km, are shown in Fig. 3. Notice that the peak Doppler velocities in *ψ* coordinates are always positioned near 90° and 270° in all radii while those peaks in *θ* coordinates move toward 180° as the radius increases. These characteristics reflect the distortion of the GBVTD geometric relationship where differences between *θ* and *ψ* increase with increasing *R.* The least square curves fit the observed Doppler velocities quite well with standard deviations of 0.8, 0.9, and 1.2 m s^{−1} on each radius. These small rms errors between the data and the least square curves indicate that resolving TC wavenumbers 0–3 accounts for the majority of the variance. The fit in Fig. 3 is better than those shown in Lee et al. (1994) using data from an airborne Doppler radar. This result is not surprising because the airborne Doppler radar data are inherently more noisy due to the uncertainties associated with a mobile platform. Wavenumber 1^{4} (i.e., primary the TC mean tangential wind) contains the majority of the variance in agreement with the findings in more intense storms such as Hurricane Gloria (1985) (Lee et al. 1994) and Hurricane Norbert (1984) (Marks et al. 1992).

The second largest component is wavenumber 0 (i.e., component of the TC mean flow along the radar–TC centerline), which contributes to the apparent asymmetry in the observed Doppler velocities. Examining Eq. (13) in Part I, the amplitude of wavenumber 0 on each radius contains not only the TC mean flow (should be a constant across all radii), but also wavenumber 1 of the TC tangential and radial winds (could be different on each radii). Therefore, the variation of the wavenumber 0 amplitude between radii is expected (Fig. 3). All other wavenumbers have amplitudes on the order of 2–3 m s^{−1}. Note that the relative importance of the different wavenumbers varies with time. For example, before Alex’s landfall, wavenumbers 2 or 3 (TC wavenumbers 1 or 2) are the second largest components (not shown).

There are two factors that may result in deviations of the Doppler velocities from the least squares fit in Fig. 3: 1) the natural variability of TC structures among different radii due to the use of a 4-km-wide annulus in collecting data for the least squares fit, and 2) small-scale circulations shown as systematic deviations of data from the curve (e.g., *θ* ≈ 270° at *R* = 35 km and *θ* ≈ 30° at *R* = 50 km in Fig. 3). Both of these factors are more pronounced due to terrain influences after Alex’s landfall. These systematic deviations could be signatures of mesocyclones or convective-scale convergence and divergence that will be addressed in a future paper.

The standard deviations (*σ*), representing the scatter of Doppler velocities along a least squares curve, can be used to assess the closeness of fit of the truncated Fourier series on the actual data. Standard deviations on each radius are computed and their mean value on each altitude (mean-std) is presented in Fig. 4a. The standard deviation of these *σ*’s on each altitude (std-std), representing the spreads of *σ*’s with respect to the mean-std, is presented in Fig. 4b. The mean-std is highly correlated with the influence of the Taiwan terrain. As Alex approached Taiwan (Fig. 2), mean-std increased and maintained a similar value after its landfall at 0517. The std-std shows a similar trend but peaks at 0517 and 4-km altitude. After Alex entered the East China Sea and moved away from Taiwan (after 0917), both mean-std and std-std decreased significantly.

The high standard deviation of individual least squares fits during the landfall stage is consistent with the influence of the terrain on the TC circulation. Organized TC circulations tend to break down (i.e., higher wavenumbers were generated) under the influence of terrain. These small-scale circulations are evident from the ragged zero Doppler velocity lines and decoupling between high reflectivity (convection) and the high Doppler velocity (primary circulation) over land (shown in the next section). In general, *σ* increases at outer radii primarily due to truncation of the Fourier series as a result of larger data gaps. Although the high wavenumber behavior of the TC circulation may not be properly represented by the truncated Fourier series, there is little effect on the quality of the retrieved mean tangential winds.

### b. Quantities derived from GBVTD-retrieved axisymmetric winds

Among the GBVTD-derived quantities, the axisymmetric tangential and radial winds have less ambiguity than their asymmetric counterparts. Between these two axisymmetric components, the tangential wind is the most robust due to its relatively large magnitude compared with the other wind components. One focus of this paper is kinematic properties of quantities derived from the axisymmetric tangential (angular momentum and perturbation pressure) and radial winds (divergence and vertical velocity). Since the axisymmetric radial winds are more sensitive to the errors in estimating TC center, only qualitative interpretation of the axisymmetric divergence and vertical velocity are justified, particularly during the landfall stage where terrain influences were at a maximum.

The axisymmetric horizontal divergence (**∇**_{H} · **V**) can be calculated from the axisymmetric radial winds (*V*_{R}):

where *R* is the TC radius and variables with an overbar represent the axisymmetric quantities (averaged around a radius). These axisymmetric quantities are functions of radius and altitude (*z*) only as defined in Part I. Vertically integrating the anelastic mass continuity equation using the axisymmetric divergence field results in the axisymmetric vertical velocity (*w*):

where *ρ* is the environmental density. In addition, *w* = 0 is applied on both the top (*z* = 10 km) and bottom (*z* = 0 km) boundaries and the O’Brien (1970) adjustment on the divergence profile is applied. The unobserved divergence in the lowest 1 km is assumed to be the same as the divergence at 1-km altitude. This assumption tends to underestimate the vertical velocity at low levels.

The unit mass axisymmetric angular momentum (*M*) in a frictionless environment with a constant Coriolis parameter (*f*) can be computed from the axisymmetric tangential velocity (*V*_{T}) as follows (e.g., Hawkins and Imbembo 1976):

The TC axisymmetric tangential winds are governed approximately by the gradient wind balance (e.g., Yanai 1964; Hawkins and Rubsam 1968; Willoughby 1979) as follows:

where *p*′ represents the axisymmetric perturbation pressure. Hence, the axisymmetric angular momentum and perturbation pressure of a TC can be computed from the GBVTD-retrieved axisymmetric tangential winds.

Since the axisymmetric tangential wind cannot be reasonably retrieved inside *R* = 5 km because few scatterers are found near the TC center, the tangential winds inside *R* = 5 km are obtained by assuming the wind speeds decrease linearly from the first available winds beyond 7-km radius to the center of the typhoon. The deviation of the wind profile inside the eye from the assumed linear profile may create some minor errors in the retrieved pressure gradient near the center. The derived axisymmetric perturbation pressure gradient is determined at each altitude using (4). The central pressure can then be calculated if the pressure values at the boundary of the analysis domain at each altitude are known. This calculation assumes that the vortex-scale pressure field is primarily due to the axisymmetric tangential wind while the asymmetric circulations produce minor perturbation upon it. Based on this assumption, the central pressure can be determined from any point in the environment provided that the edge of the analysis domain is close to the “environment.” Since the exact pressure on the boundary is usually unknown, we cannot retrieve the absolute pressure but the pressure gradient at each altitude (Gal-Chen 1978).

### c. Operation limits of the GBVTD technique

Theoretically, the GBVTD technique is limited only by the geometric distortion, which is measured by sin*α*_{max}. It has been shown in Part I that the tolerable upper limit on sin*α*_{max} is wavenumber dependent (i.e., inversely proportional to the wavenumber).For example, the upper limit of sin*α*_{max} varies from 1 for resolving TC wavenumber 0 to 0.33 when TC wavenumber 3 is included. Since sin*α*_{max} is always <1.0, this implies that the largest radius suitable for GBVTD analysis cannot be larger than *R*_{T}. This criterion effectively shrinks the analysis domain as a TC moves toward a Doppler radar.

When applying the GBVTD technique to real TCs, both the radar characteristics (e.g., beamwidth, pulse repetition frequency, etc.) and the storm characteristics (e.g., vertical extent, RMW, etc.) determine the maximum range that GBVTD technique can be applied. Detailed discussions of these factors are provided in the appendix for interested readers. The best range for the GBVTD technique to resolve the storm-scale circulation (with an average beam resolution of 2 km) is when the TC is located between 60 and 150 km from a WSR-88D. The average separation of the current WSR-88D network is ≈250 km. Hence, a TC moving out of the best range for one WSR-88D will likely enter the best range of a nearby WSR-88D, making continuous monitoring of a TC circulation using the GBVTD technique feasible. However, gaps in GBVTD coverage are unavoidable when a WSR-88D radar is inside the RMW of a TC.

## 4. Storm evolution and asymmetric structure

The evolution of Alex’s vortex is illustrated using 2-km altitude CAPPI reflectivity and Doppler velocity and the GBVTD-derived relative tangential winds. The relative tangential wind is defined as the sum of all wavenumbers except for the mean flow, whereas the total tangential wind is defined as the sum of the relative tangential wind plus the mean flow. Note that the mean flow deduced by the GBVTD analysis is the component of the mean flow along the radar–TC center direction, defined in Part I. Hence, the total tangential wind is an approximation of the “full tangential wind” when the data from only one radar are available. In this study, the relative tangential wind is used in the discussion unless otherwise specified. For the convenience of discussion, the 16 time periods of data are divided into three stages: prelandfall stage (0432–0502), landfall stage (0517–0647), and postlandfall stage (0917–1032). Landfall is defined as the time when the TC center crosses the coastline.

### a. Prelandfall stage

At 0432, Alex’s entire eyewall was within CAA’s radar range (Fig. 5a). The maximum inbound Doppler velocity exceeded 45 m s^{−1} in the northern portion of the eyewall while the maximum outbound velocity exceeded 20 m s^{−1} just onshore the southwestern eyewall ≈(60 km, −60 km) (hereafter, all coordinates are in units of km). The eyewall at 0432 was composed of two distinct reflectivity bands (labeled A and B in Fig. 5a), which are typical in weaker TCs (Willoughby et al. 1982). Bands A and B also denote two separate regions of Doppler velocity maximum exceeding 45 m s^{−1} SE of the center (not shown in Fig. 5a due to contour interval selection). Another rainband labeled C extended over the western portion of the eyewall and was prominent at 5-km altitude (not shown). These bands consolidated between 0432 and 0447, where the upwind edge of band B merged with band A, and were related to a single inbound velocity maximum >50 m s^{−1} [at (80, −21) in Fig. 5c]. The eyewall was comma shaped with rainbands spiraling out to the east. These rainbands rotated counterclockwise around the circulation center and the upwind ends of band A, B, and C could still be identified in Fig. 5c at 0447. The radar reflectivity pattern suggested that other spiral bands existed on the east and southeast quadrants of the eyewall. By 0503, rainbands A and B had merged into rainband C and formed a large spiral band.

The low-level tangential winds at 0432 (Fig. 5b) indicated two maxima in the southeast and northeast quadrants of the TC. In general, the eyewall reflectivity maximum was associated with high winds. There were some discontinuities in the tangential wind field near *R* = 30 km owing to the data void beyond the unambiguous range of the radar, an obvious limitation of the GBVTD analysis when only part of the storm was within the effective Doppler range. As Alex moved closer to the radar, the GBVTD-retrieved wind fields became more coherent. The tangential winds at 0447 (Fig. 5d) showed a comma-shaped pattern, in good agreement with the eyewall reflectivity with the upwind portion wrapping around east of the center. Maximum tangential winds at 0447 were located north-northwest of the typhoon center with 51 m s^{−1} maxima at 2-km altitude (Fig. 5d). This51 m s^{−1} maximum (located at *R*= 20 km and 338° azimuth) was the sum of a mean tangential wind at 38.9 m s^{−1}, a wavenumber 1 of 9.1 m s^{−1}, a wavenumber 2 of −2.0 m s^{−1}, and a wavenumber 3 of 4.9 m s^{−1}. At this location, the angle between the radar beam and the tangential velocity vector was 50° because the tangential wind was along 248° while the radar viewed along 300°/120°. Hence, the 51 m s^{−1} tangential wind produced a 33 m s^{−1} Doppler radial velocity toward the radar, which, when added to the 8 m s^{−1} mean flow toward the radar, yeilds the observed 41 m s^{−1} Doppler velocity toward the radar in Fig. 5c. While the retreived wind is subject to aliasing errors from the unresolved asymmetric radial flow (a limitation/assumption of the GBVTD technique), the major strength of the GBVTD technique is the ability to use the gradient of the Doppler velocity along a constant radius from the TC center to retreive the winds without observing the actual maximum and minimum.

Typhoon Alex reached its peak intensity at 0447 shortly before it made landfall, as defined by the occurrence of the peak inbound Doppler velocity and the tightest Doppler velocity gradient north of the center in Fig. 5c. Achieving this level of detail in the GBVTD-retrieved tangential winds represents a big step forward compared with the structures inferred from the single-Doppler velocity patterns. Alex’s intensity weakened after the eyewall reflectivity impinged on the terrain in northeastern Taiwan around 0503 where the maximum wind in the eyewall (NW of the eye) dropped below 50 m s^{−1}. The 50 m s^{−1} wind maximum ≈10 km south of the center seemed questionable. In fact, this wind maximum is caused by a large (≈18 m s^{−1}) wavenumber 1 maximum inside the RMW superimposed on a ≈30 m s^{−1} axisymmetric tangential wind. This unusually large wavenumber 1 component in the eye can be a result of multiple vortices forming inside RMW when the eyewall impinges on the terrain. When the GBVTD-simplex derived TC circulation center is optimized for the primary eyewall around *R* = 23 km, this center may not be the center for the small-scale vortices near the TC center. The zigzag of the centers at 0447 and 0502 with respect to the mean track may reflect the uncertainty in center locations due to multiple vortices. More evidence of these multiple vortices will be discussed in the next subsection.

### b. Landfall stage

Alex’s center crossed the Taiwan coastline around 0517 (not shown). This stage is characterized by slow disintegration of the organized eyewall circulation due to the influence of terrain. After 0517, the eye began to fill with small, shallow convective cells (echo top ⩽6 km, not shown) similar to those documented in Muramatsu (1986). At 0533 (Fig. 6a), a small ring of high reflectivity [≈18 km diameter centered at (67, −27), referred to as the inner eyewall] formed within and attached to the southern part of the original eyewall. A similar but weaker reflectivity structure was visible at 0517 connected to the western part of the original eyewall (not shown). Assuming this feature moved with the mean tangential wind of ≈35 m s^{−1} at 23-km radius, it would rotate a quarter circle in ≈17 min, consistent with the location of the feature at 0533. Unfortunately, the 15-min period between volume scans does not allow us to trace the origin of this inner eyewall before 0517.

Vertical cross sections (at 0533) through the center of the inner eyewall are shown in Fig. 7. The east–west cross section (Fig. 7a) indicated that the reflectivity on the west side (*x* ≈ 60 km) is higher than that on the east side (*x* ≈ 73 km). The pattern of Doppler velocities indicated that there is convergence (divergence) near the west (east) side of the inner eyewall, a feature that can be inferred from the curvature of the zero Doppler velocity line through the inner eyewall (Fig. 6a) illustrated in Fig. 4 in Part I. Note that the Doppler velocities in Fig. 7b are in and out of the page; therefore, they show a counterclockwise rotation. The original and inner eyewalls did not share the same circulation center (Fig. 6a) as evidenced by the offset of the typhoon symbol and the center of the inner eyewall. In fact, the GBVTD-simplex algorithm did identify a center at (61.5, −32.3) and 2-km altitude with an RMW of 9 km and a maximum axisymmetric tangential wind of 18 m s^{−1}. This offset demonstrates how the single vortex center assumption used by the GBVTD analysis can break down in complex flow.

The filling of Alex’s eye accelerated after 0533 and the inner eyewall cannot be identified at 0547 (not shown). As Alex moved farther inland, its intensity decreased rapidly as indicated by the diminishing inbound Doppler velocities with time. Tracing the motion of the minimum radar reflectivity from 0517 to 0632 (left panels in Fig. 6) suggests it moved around the terrain paralleling Taiwan’s northeastern coast. Comparison of the location of the reflectivity minimum and the circulation center deduced from the GBVTD-simplex algorithm, or inferred from the zero Doppler velocity line, suggested that this reflectivity minimum was displaced ≈20 km to the north of the circulation center. From 0602 to 0632, the inbound maximum Doppler velocities were collocated with the reflectivity minimum while the circulation center was associated with high reflectivity. This reflectivity–velocity decoupling is not commonly observed in mature TCs over the ocean and is likely a sign of terrain influence. The magnitude of the inbound Doppler velocity maximum decreased with time while the outbound velocity maximum associated with convection over land increased with time. Stationary reflectivity features on the western slopes of the central mountain range [e.g., radar reflectivity maximum at (25.0, −20.0)] were also observed during this period.

The GBVTD-derived relative tangential winds from 0532 to 0632 are shown in the right panels of Fig. 6. As Alex approached the CAA radar, the effective radius of the GBVTD analysis reduced from 60 km at 0532 to 48 km at 0602 and finally to 40 km at 0632. This shrinking analysis domain results from limiting sin*α*_{max} < 0.9 to minimize the geometric distortion discussed in Part I. During this time, the peak relative tangential wind speed dropped from 45 m s^{−1} at 0517 to 37 m s^{−1} at 0632. The overall wind pattern within the eyewall possessed numerous wind maxima but in general the highest wind speed was on the east side of the storm between 0532 and 0617. This portion of the TC was mostly over the ocean and least affected by the terrain. A secondary maximum, on the west side of the TC, was associated with enhanced reflectivity over land.

### c. Postlandfall stage

After the CAA site regained power at 0917, Alex was located 60 km north of the radar and tracked NNW for the next few hours. The zero Doppler velocity line curved westward beyond the typhoon center and the centers were located east of the zero Doppler velocity line (e.g., 0932 and 1004; Figs. 8a and 8c). From these two signatures we can infer that the mean flow was southerly (see Part I), in good agreement with the 6–7 m s^{−1} GBVTD-retrieved southerly mean flow *V*_{M} cos(*θ*_{T} − *θ*_{M}) (Table 1). Since the typhoon motion and the radar viewing angle are almost parallel to each other, *θ*_{T} ≈ *θ*_{M} and *V*_{M} cos(*θ*_{T} − *θ*_{M}) ≈ *V*_{M}.

At 0932 (Fig. 8a), most of the high reflectivity was located in the southeast quadrant and connected to precipitation over northern Taiwan. Another rainband was located north of the center. Scattered reflectivity maxima can be seen surrounding the TC center indicating residual reflectivity inside the eyewall from Alex’s landfall. A ring of high reflectivity began to form outside the RMW at 1004 (Fig. 8c) and can be seen until 1047 (not shown). The peak outbound and inbound Doppler velocities remained ≈35 and ≈−25 m s^{−1} from 0917 to 1047 with similar patterns throughout this period. This trend suggests that Alex attempted to redevelop after leaving the terrain influence of Taiwan.

The GBVTD-derived relative tangential winds are illustrated in the right panels of Fig. 8. The relative tangential wind is characterized by a wavenumber 1 component located on the north (front) side of the storm. From 0917 to 1032, the relative tangential winds at 1 km are generally stronger on the west side of the center (not shown). These characteristics agree with simulations of axisymmetric and slow-moving hurricanes (Shapiro 1983) where the enhanced inflow and convergence in front of the hurricane are caused by friction in the boundary layer. The reflectivity maximum and the Doppler velocity on the north and northwest quadrants of the TC intensified and broadened, reflected as the widening of the 32 m s^{−1} contour northwest of the center at 1004 and 1032.

When adding mean flow to the relative tangential wind field, the tangential winds have a maximum on the northeast quadrant of the storm (not shown), which is consistent with the Doppler velocity pattern (Fig. 8) and in agreement with previous observations (e.g., Gray and Shea 1973). Therefore, the apparent asymmetry in the Doppler velocity pattern is primarily due to the mean flow instead of the wavenumber 1 asymmetry, consistent with the orientation of the zero Doppler velocity contour illustrated in Part I. More importantly, the GBVTD analysis revealed a wavenumber 1 asymmetry north of the center that cannot be inferred from the Doppler velocity pattern.

## 5. Axisymmetric circulation

In this section, Typhoon Alex’s circulation is discussed using the axisymmetric tangential winds, radial winds, and their derived divergence, vertical velocity, angular momentum, and perturbation pressure. These quantities were computed on a 1-km grid in both radius and height, representing the axisymmetric mean fields.

### a. The prelandfall stage

The axisymmetric vortex of Typhoon Alex at 0447 is illustrated in Fig. 9 where the three components of the velocity field are shown in panels (a) and (b) and the angular momentum and perturbation pressure fields are shown in panels (c) and (d). At 0447, the axisymmetric tangential wind maximum reached ≈39 m s^{−1} and the RMW was 22 km. Alex’s maximum axisymmetric tangential wind was about 14 m s^{−1} weaker than that of Hurricane Norbert, the weakest of the three hurricanes studied previously. There was a deep (≈2 km) layer of inflow^{5} at low levels (Fig. 9b), a layer of outflow in the midlevel, and inflow at the upper level sloping down along the inner edge of the eyewall toward the center. The low-level inflow (≈−7 m s^{−1}) peaked at *R* = 27 km and *z* = 1 km, and the convergence zone extended inward to the inner edge of the eyewall at *R* = 15 km. The primary updraft (≈1.5 m s^{−1}) coincided with the maximum tangential wind. When the updraft encountered the downdraft at high levels (e.g., *R* = 25 km, *z* = 7.5 km), part of the updraft entered the eye and the rest merged with the downdraft and flowed outward at midlevels. Downdrafts were resolved inside the sloping eyewall reflectivity. Again, although the patterns and magnitudes look plausible, some unrealistically large vertical velocities were generated by questionable radial wind gradients inside the eye above *z* = 6 km, which is a problem when few scatters are available.

The angular momentum (Fig. 9c) shows a similar profile to that in Hurricane Norbert (Marks et al. 1992) where the contours are nearly vertical inside the eyewall (*R* = 15 km) and slope outward with height. If an air parcel conserves angular momentum in a frictionless environment ignoring the Coriolis effect, the slope of constant angular momentum surface is a function of the maximum axisymmetric tangential wind, the RMW, and the vertical shear of the axisymmetric tangential wind at the RMW [eq. (4) in Jorgensen (1984)]. The estimated slope of the constant angular momentum surface in the eyewall at 0447 is 28° from the horizontal. This constant angular momentum slope is consistent with a 25° slope computed from the eyewall radar reflectivity (e.g., 25-dB*Z* contour), but not as steep as the 60° reported in other hurricanes over the ocean (e.g., Marks and Houze 1987; Marks et al. 1992; Lee et al. 1994). However, a similar slope of eyewall reflectivity was reported in the much stronger Hurricane Allen (1980) by Jorgensen (1984). These results suggest that the eyewall slope is not entirely determined by the storm intensity.

Generally, the angular momentum increases with radius because the mean tangential wind decreases more slowly than the radius increases. The maximum angular momentum of 16 × 10^{5} m^{2} s^{−1} was located at *R* = 60 km and *z* = 6 km due to the upward and outward slope of the peak axisymmetric tangential wind. In a layer below *z* = 2 km and beyond *R* = 20 km, the angular momentum contours slope radially outward with height, which is an indication of angular momentum loss at the bottom of the inflow layer due to friction (e.g., Marks et al. 1992). This angular momentum loss may be partially compensated by the boundary layer inflow, which crosses the constant angular momentum contours, transporting higher angular momentum inward.

The perturbation pressure field is shown in Fig. 9d. This bell-shaped pressure deficit pattern is similar to that in a simulated axisymmetric hurricane (Fig. 5b in Rotunno and Emanuel 1987). The nearly vertical pressure contours (e.g., −3- and −6-mb contour lines) below 5 km are due to the outward and upward slope of the tangential wind maximum. The horizontal pressure gradient maximum is located at *z* = 1 km and decreases with height. There is less than 3-mb horizontal pressure drop at *z* = 10 km compared with a 21-mb pressure drop at *z* = 1 km across the 60-km domain. The vertical perturbation pressure gradient is strongest at the TC center and decreases as the radius increases. The downdraft inside the eye is downgradient, but the low-level outflow from the typhoon center is countergradient. The existence and maintenance of this outflow from the TC center requires downward motion inside the eyewall, which is consistent with the flow pattern resolved here.

### b. Landfall stage

The axisymmetric structure at 0533, illustrated in Fig. 10, changed significantly during the 45 min since 0447. The eye was filled with high reflectivity and the low-level inflow penetrated all the way to the typhoon center (Fig. 10b). The outward tilting eyewall reflectivity at 0447 was replaced by a widespread reflectivity pattern, stratiform in appearance. The axisymmetric tangential winds at 1-km altitude (Fig. 10a) decreased rapidly at all radii indicative of increasing surface friction as Alex moved over land. The unrealistically large inflow above *z* = 7 km and beyond *R* = 30 km corresponds to poor data coverage at high altitude and large radii. As Alex moved farther inland, the height of the maximum tangential winds rose from 1- to 3-km altitude after 0602 and increased radially outward (not shown), consistent with the increasing terrain influence at low levels.

In response to the evolution of the axisymmetric tangential wind, the low-level angular momentum (Fig. 10c) decreased beyond the RMW (*R* = 30 km) while the angular momentum at midlevel increased. This change in structure is consistent with decaying low-level storm intensity after Alex’s landfall while the midlevel wind speed actually increased slightly after Alex’s landfall. The perturbation pressure pattern (Fig. 10d) showed that the pressure deficit at *z* = 1 km decreased from −20 mb at 0502, to −16 mb at 0533, and finally to −11 mb at 0632.

### c. Postlandfall stage

After Alex left Taiwan, its circulation became nearly axisymmetric with the RMW located at ≈30 km. High reflectivity filled the eye and the maximum reflectivity was stronger than that during the landfall stage. An example of axisymmetric structure at 1032 is shown in Fig. 11. Enhanced eyewall reflectivity coincided with the peak tangential wind around *R* = 30 km. The area of strong tangential wind (e.g., 25 m s^{−1} contour) is shallow and spread horizontally through a broad radial band that does not resemble the upright structure shown in mature tropical cyclones (e.g., Marks and Houze 1987; Marks et al. 1992; Lee et al. 1994).

The radial flow (Fig. 11b) showed an inflow at 1-km altitude beyond 44-km radius and suggested inflow below 1 km penetrated to the RMW. There was mid- to upper-level outflow beyond *R* = 35 km. A reflectivity maximum at *R* = 40 km intensified with time, consistent with the convergence at *R* = 44 km and 1-km altitude, and the updraft above the convergence. The very weak updraft associated with the primary eyewall (*R* = 28 km) is probably due to the underestimated convergence below 1-km altitude. The existence of the outer secondary circulation may reduce the supply of the high angular momentum air reaching the inner reflectivity maximum. The double reflectivity and wind maxima structure resembled many aspects of an eyewall replacement cycle discussed in Shapiro and Willoughby (1982) and Willoughby et al. (1982). Some indication of the double-eyewall structure can also be found at 1032 (Fig. 8e).

The angular momentum and perturbation pressure increased slightly compared with those in the landfall stage suggesting that Alex reintensified after it left the influence of Taiwan’s terrain. Although the maximum tangential wind decreased slightly after 1004 (not shown), the widening of the tangential wind maximum produced an additional 2-mb pressure deficit at 1047 (not shown).

## 6. Comparison of the retrieved perturbation pressure with surface pressure observations

There was no aircraft reconnaissance in Typhoon Alex. The surface wind, highly affected by the terrain, did not provide a good direct comparison with the retrieved wind field at *z* = 1 km. However, there were five surface stations reporting pressure at 0500 and 0600 in northeast Taiwan within 70 km of the typhoon center (Fig. 12a). As a further check of the GBVTD-derived winds, we compare the surface pressure gradient computed from these surface pressure reports with the retrieved surface perturbation pressure gradient.

The GBVTD-retrieved surface perturbation pressure gradient at different radii from the typhoon center was estimated by extrapolating the pressure fields retrieved at 0502 and 0602 downward from *z* = 1 km. Additionally, we computed the likely uncertainty of the retrieved pressure field by imposing a ±2 m s^{−1} bias (rms of the Doppler velocity about the GBVTD curve presented in Fig. 4 at 1-km altitude for 0502 and 0602) to the axisymmetric tangential winds across all radii. The envelope of the uncertainty expands toward the TC center where a ±3 mb accumulated uncertainty is found at the TC center. The size of the envelope depends on the uncertainty in the GBVTD-retrieved axisymmetric tangential winds. With a smaller uncertainty in the postlandfall stage, the retrieved pressure gradient will be more accurate than the example shown here. In addition, the uncertainty in the pressure estimates will be much smaller if the errors in the axisymmetric tangential winds are random rather than a bias.

The surface pressure observed at each station was subtracted from the pressure at the station farthest from the typhoon center (690 at 0500 and 706 at 0600) to obtain the “observed” axisymmetric surface pressure gradient (Fig. 12b). In this example, the central pressure of Alex, ≈969.5 mb (973.8 mb) at 0502 (0602), can be estimated by adding the pressure deficit, ≈−23 mb (−16 mb) from *R* = 60 km (48 km), to the measured surface pressure of 992.5 mb (989.8 mb) at station 690 (706). Although there were no surface pressure observations within 20 km of Alex’s center, the “GBVTD-retrieved”pressure gradient beyond 20-km radius was consistent with the observed surface pressure gradient. Considering all the assumptions (e.g., axisymmetry, gradient wind approximation, uncertainty in TC center, etc.) involved in retrieving the perturbation pressure, and the influence of the terrain on each station, the 1–2-mb deviations between the retrieved and the observed surface pressure gradient is remarkably small. This agreement not only indicates the consistency of the GBVTD-retrieved axisymmetric circulation in Typhoon Alex, but also suggests that our technique has the potential to retrieve the central pressure of a TC within 2–3 mb. Therefore, it may be feasible to estimate central pressure of a landfalling TC from the GBVTD-retrieved axisymmetric tangential winds in conjunction with one or two surface pressure measurements within the GBVTD-analysis domain. More case studies will be performed and compared with surface and aircraft dropsonde data in the future.

## 7. Summary and conclusions

This paper completes a three-part study of a ground-based single-Doppler radar TC wind retrieval technique, GBVTD, by presenting applications of the GBVTD technique on landfalling Typhoon Alex. Under different storm intensity and degrees of terrain influence, we demonstrate that plausible and physically consistent primary circulations of a TC can be retrieved from ground-based single-Doppler radar data using this technique and the GBVTD-simplex center finding algorithm. In addition, the derived axisymmetric dynamical parameters (angular momentum and perturbation pressure) are internally consistent with the radar reflectivity structures.

A total of 16 volumes (6.5 h data with a 2-h gap) were collected while Typhoon Alex moved across the mountainous terrain of northern Taiwan. Before landfall, Alex was characterized by a highly asymmetric circulation with a maximum axisymmetric tangential wind of 39 m s^{−1} and a maximum total tangential wind of ≈52 m s^{−1} at *z* = 2 km, north-northwest of the center. Also, the eyewall reflectivity was tilted ≈25° from horizontal in agreement with the slope of the constant angular momentum surface within the eyewall reflectivity. Alex reached its minimum surface pressure, ≈969 mb (a 23-mb pressure deficit from *R* = 60 km), at 0447. A deep layer of low-level inflow collided with outflow from the eye at the bottom of the eyewall and produced a broad region of updraft. This inflow also brought in higher angular momentum air from outer radius into the eyewall region.

Alex’s circulation weakened after landfall and the eye began to fill. The low-level axisymmetric wind maximum gradually elevated from 1- to 3-km altitude and expanded outward. The reflectivity center (minimum reflectivity) and circulation center (maximum vorticity) were decoupled as Alex moved farther inland. The circulation center moved across the terrain while the reflectivity center moved around the terrain following the coastline. Within 90 min after landfall, the retrieved central pressure filled ≈10 mb. Upon leaving Taiwan’s northern coast, Alex attempted to reintensify as the central pressure deepened 3–4 mb in the next 2 h. The maximum axisymmetric winds reappeared at 1-km altitude and broadened horizontally with the eyewall reflectivity maximum.

The retrieved perturbation pressure fields at 0502 and 0602 were compared with the surface pressure reported over northern Taiwan. The observed and GBVTD-retrieved pressure gradient was within only 1–2 mb beyond 20-km radius. The agreement seems very good considering the assumptions involved and the influence of terrain. This agreement suggests that the GBVTD-retrieved quantities can be used for operational and research purposes.

The simple circular vortex and single circulation center assumptions in the GBVTD technique and GBVTD-simplex algorithm caused some problems in Typhoon Alex when multiple vortices formed as the primary circulation impinged on the terrain. Nevertheless, while the break down of these assumptions has some impact on the asymmetric circulation, it has little effect on the axisymmetric tangential wind.

In conclusion, the GBVTD technique provides a new way to examine TC circulations from single ground-based Doppler radar. The ability to resolve realistic 3D axisymmetric and asymmetric structures of a TC near landfall in real time not only expands the capability of using ground-based Doppler radar data in TC forecasts but provides researchers an opportunity to examine TC kinematic and some derived dynamic structures. There is no doubt that the GBVTD technique needs to be tested on a wider spectrum of TCs in conjunction with other independent measurements so a more complete comparison of the results can be performed. A version of the GBVTD analysis package that uses coarser-resolution WSR-88D level IV data has been implemented in near–real time at the National Hurricane Center in Miami to provide low-level wind structure in landfalling TCs to forecasters. We plan to perform GBVTD analysis on Hurricane Danny (1997) and Georges (1998) where a more rigors comparison can be performed with aircraft in situ measurements, dropsonde data, and dual-Doppler analysis.

## Acknowledgments

The authors thank the CAA for providing the Doppler radar data. The authors are grateful to Drs. Peter Hildebrand, Tammy Weckwerth, Mr. Peter Dodge, Mr. Vincent Wood, Mr. Scott Ellis, and two anonymous reviewers for their valuable comments that greatly improved this paper. Ms. Susan Stringer helped prepare figures and Ms. Jennifer Delaurant proofread the manuscript. This research is supported by the National Science Foundation, the National Oceanographic and Atmospheric Administration, the National Science Council of Taiwan, Republic of China (ROC), and the Technology Advisors Office, Ministry of Transportation and Communication of Taiwan, ROC, under Grant NSC89-2111-M002-019-Ap6.

## REFERENCES

### APPENDIX

#### Practical Considerations of the GBVTD Technique

The operational limits and accuracy of the GBVTD-derived TC circulations are constrained by the earth’s curvature, radar characteristics, beam geometry, scanning strategy, and the uncertainties in vertical velocity and precipitation terminal velocity. Here, these factors are discussed to better understand and interpret the GBVTD-derived winds in real TCs. Each factor provides its own limits to the GBVTD technique.

##### Earth curvature

The propagation path of a radar beam is affected by the refractive index and the altitude of a horizontally pointing beam, which increases with distance from the radar due to the curvature of the earth (Rinehart 1991, chapter 3, Fig. 3.2). The mean sea level altitude at the center of a radar beam at various distances are shown in Fig. A1. The radar altitude needs to be added to Fig. A1 to obtain the true beam altitude. The altitude of the lowest elevation angle and the vertical extent of the TC’s eyewall determine the maximum distance a TC center can be detected by a radar. The center of the 0.5° beam exceeds 1-km altitude beyond 80 km from the radar. As a result, it is possible to resolve a limited portion of circulation beneath 1-km altitude when the TC center is within 60 km of the radar. For a typical TC with the eyewall convection reaching 10-km altitude, the TC center can be detected by a ground-based radar (at 0.5° elevation angle scan) up to *R* = 340 km. However, the maximum effective range of the GBVTD technique is further restricted by other factors discussed in this appendix.

##### Scanning strategy

For a given TC, the solid angle required for a radar to sample the same region of the storm (for a given diameter and altitude) increases as the storm approaches the radar. This effect results in the need for a larger azimuth angle and elevation angle sector to be scanned by a radar. The effect of a larger azimuth angle sector (equivalent to increasing *α*_{max}) increases the contribution from the unknown cross-beam component of the mean flow into the mean tangential winds and limits the ability to recover the higher wavenumber wind components due to distorted geometry (see Part I).

For a TC with the top of the eyewall convection at 10 km and a 30-km-radius eyewall, a Doppler radar needs to scan up to 9° elevation when the center of the storm is located at 150 km from the radar in order to cover the eyewall at 120 km (see Fig. A1). A radar needs to scan up to the 20° elevation when the TC center is 100 km from the radar. As a result, the errors in particle terminal fall speed (*υ*_{t} estimation and unknown vertical velocity, *w*) could contaminate the results at closer range (discussed later in this appendix). As the elevation angle increases, an operational Doppler radar has to scan in larger elevation angle steps (i.e., sacrificing vertical resolution) to complete a volume scan within a preset time interval.

##### Beam spreading

The beamwidth of a Doppler weather radar is usually between 0.5° and 1°. A typical 1° beam is ≈7.5 km wide at 400-km range from the radar accounting for two-thirds of the depth of a typical TC. To sample a TC with less than a 3-km-average beamwidth, the storm center must be within 180 km of a Doppler radar with a 1° beam. At larger ranges, the magnitude of TC tangential wind maximum will be smoothed by the beam.

##### Range and velocity ambiguity

The maximum unambiguous range and velocity of a pulsed Doppler radar transmitting at a single pulse repetition frequency (PRF) are governed by the following equations (Doviak and Zrnic 1993):

where *λ, C,* PRF are the wavelength of the radar (m), speed of electromagnetic wave (3 × 10^{8} m s^{−1}), and pulse repetition frequency (s^{−1}), respectively. It is clear that *R*_{max} depends only on PRF while *V*_{max} depends on both PRF and *λ. *Table A1 shows the unambiguous range and Doppler velocity at typical PRFs for C- and S-band Doppler radars.

The maximum winds of a TC range from 34 up to 80 m s^{−1} or higher. In order to minimize velocity aliasing, *V*_{max} > 20 m s^{−1} is preferred, further limiting the effective range to <200 km. Implementation of a dual-PRF mode effectively increases *V _{N}* (Frush 1991; Loew and Walther 1995) as

where PRT = 1/PRF is the pulse repetition time (s) and PRT_{2} > PRT_{1}. However, *R*_{max} is still limited by the higher PRF of the two.

##### The uncertainties in terminal and vertical velocities

The precipitation particle terminal velocity and air vertical velocity contribute to the Doppler velocity when the elevation angle of a beam is nonzero [Eq. (2) in Part I). The terminal velocity (*υ*_{t}) can be estimated using reflectivity factor (*Z*) and *υ*_{t} relationships such as

where (A4) is for rain (Joss and Waldvogel 1970), applied to altitudes <5 km, while (A5) is for snow (Atlas et al. 1973), applied to altitudes >7 km. The relationship applied to altitudes from 5 to 7 km is obtained from a linear interpolation of the above two relationships. The density correction factor, exp(−*H*/9.58)^{0.4}, is from Beard (1985) where *H* is the altitude (km). At 5-km altitude, *Z* = 50 dB*Z* gives *υ*_{t} = 10 m s^{−1}. The projection of the *υ*_{T} on Doppler velocity (*V*_{d}) at an elevation angle of 15° is ⩽2 m s^{−1}. Even with a 50% uncertainty in estimating particle terminal velocity, the error in the Doppler velocity from the terminal velocity uncertainty is ⩽1 m s^{−1}.

Intense vertical velocities (≥10 m s^{−1}) within a TC are usually located in the eyewall and outer rainbands (Black et al. 1996). As discussed in Lee et al. (1994), ignoring *w* in Eq. (2) in Part I produces ⩽2 m s^{−1} uncertainty in *V*_{d} even in the eyewall region. Hence, the errors due to uncertainties in *w* and *υ*_{t} are small (usually ⩽10% because they tend to partly cancel each other) compared with *V*_{T} (≥34 m s^{−1}) of a TC at hurricane strength.

## Footnotes

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

+ Current affiliation: Wu-Fen-Shan Radar Station, Central Weather Bureau, Taipei, Taiwan, Republic of China.

*Corresponding author address:* Dr. Wen-Chau Lee, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000.

Email: wenchau@ucar.edu

^{1}

The three components of the TC circulation in the TC cylindrical coordinates are termed tangential wind, radial wind, and vertical wind, while the single-Doppler velocity is termed Doppler velocity.

^{2}

In a circular flow model, the most dynamically consistent TC circulation center is a point that maximizes the axisymmetric tangential wind at the radius of maximum wind (Part II).

^{3}

The center is defined as the average position among all possible centers (within one standard deviation) computed from various initial guesses. The uncertainty is estimated from the standard deviation among all runs.

^{4}

The wavenumber *n* in the least squares curve fit represents primarily the wavenumber *n* − 1 in TC circulation due to the projection of the TC circulation on to the radar beam direction. See Part I for details.

^{5}

Note the data below *z* = 1 km are not observed by CAA radar. However, it is reasonable to assume inflow exists below *z* = 1 km by extrapolating the coherent flow structure downward and the magnitude of the inflow may even be stronger near the surface.