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  • View in gallery

    The geometry and symbols used in the formulation of the GBVTD: T is the vortex center; E is the observation point; Vd/cosϕ is the horizontal projection of the observed Doppler velocity; and VM, VT, and VR are the mean wind, tangential wind, and radial wind of the vortex, respectively (adapted from Lee et al. 1999).

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    (left) [a(1)] A constant easterly mean wind with a magnitude of 10 m s−1, [b(1)] a Rankine-combined vortex, [c(1)] the axisymmetric radial wind, and [d(1)] the total wind [(a1) + (a2) + (a3)], and the corresponding (middle) observed Doppler velocity and (right) VdD/RT display of the simulated wind fields in the left-hand side. The Doppler radar is located at the lower-left corner.

  • View in gallery

    A comparison of Vd and VdD/RT displays for two different radii (delineated as red and blue lines) for two vortices with different RMWs: (a) Vd display for a pure rotating vortex with Rmax = 30 km; (b) Vd profiles at R = 30 and 60 km; (c) same as (a), but for VdD/RT display; and (d) same as (b), but for VdD/RT profiles. (e)–(h) Same as (a)–(d), but for Rmax = 80 km > RT and the two Vd profiles are at R = 80 and 110 km. The center “T” is located at (x, y) = (200 km, 200 km) and the hypothetical Doppler radar “O” is located at (x, y) = (150 km, 150 km), with RT = 50√2 km.

  • View in gallery

    Comparison of GBVTD- and GVTD-retrieved vortex structure. [a(1)] The simulated axisymmetric wind field, [a(2)] the GBVTD-retrieved wind field, and [a(3)] the GVTD-retrieved wind field. Same as [a(1)]–[a(3)], but for [b(1)]–[b(3)] wavenumber one, [c(1)–[c(3)] wavenumber two, and [d(1)–[d(3)] wavenumber three cases. The center is located at (0, 0), while the hypothetical Doppler radar is located at (0, −80).

  • View in gallery

    Percentage error distribution of the retrieved vortex as a function of θ′ (x axis, °) and R (y axis, km) for (a) wavenumber two GBVTD-retrieved VT, (b) wavenumber three GBVTD-retrieved VT, (c) wavenumber two GVTD-retrieved VT, and (d) wavenumber three GVTD-retrieved VT.

  • View in gallery

    Comparison of GBVTD- and GVTD-retrieved pure rotational vortex structure with prescribed center displacement along the RT vector (y axis). Results are for the case of [a(1)] GBVTD- and [a(2)] GVTD-retrieved structure for a center displacement of 1 km. Same as [a(1)] and [a(2)], but for a center displacement of [b(1)], [b(2)] 5 and [c(1)], [c(2)] 10 km, respectively.

  • View in gallery

    Percentage error distribution of the retrieved vortex as a function of θ′ and R. Results are for the case of (a) GBVTD- and (b) GVTD-retrieved vortex with a center displacement of 5 km; (c), (d) same as (a), (b), but for a center displacement of 10 km.

  • View in gallery

    RMSE distributions of the retrieved VTmax and VRmax from GBVTD and GVTD corresponding to center misplacement as a function of x and y from the true center: VTmax retrieved by (a) GBVTD and (b) GVTD, and VRmax retrieved by (c) GBVTD and (d) GVTD.

  • View in gallery

    Effect of mean wind on the vortex retrieved by GVTD. RMSE of (a) retrieved VTmax and (b) retrieved VRmax. The mean wind is perpendicular to the RT vector with a magnitude of 10 m s−1.

  • View in gallery

    The (a) Vd and (b) VdD/RT CAPPI displays of Typhoon Gladys at 4-km altitude observed by the Civil Aeronautical Administration Doppler radar at northern Taiwan. Simulated (c) Vd and (d) VdD/RT derived from the parameters VM = 20 m s−1, θM = −90° (easterly), VTmax = 35 m s−1, and VR = 0.

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Generalized VTD Retrieval of Atmospheric Vortex Kinematic Structure. Part I: Formulation and Error Analysis

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  • 1 Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan
  • 2 National Center for Atmospheric Research,* Boulder, Colorado
  • 3 Department of Atmospheric Sciences, National Taiwan University, Taipei, Taiwan
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Abstract

The primary circulation of atmospheric vortices, such as tropical cyclones and tornadoes, can be estimated from single-Doppler radar observations using the ground-based velocity track display (GBVTD) algorithm. The GBVTD algorithm has limitations in the following four areas: 1) distortion in the retrieved asymmetric wind fields, 2) a limited analysis domain, 3) the inability to resolve the cross-beam component of the mean wind, and 4) the inability to separate the asymmetric tangential and radial winds. This paper presents the generalized velocity track display (GVTD) algorithm, which eliminates the first two limitations inherent in the GBVTD technique and demonstrates the possibility of subjectively estimating the mean wind vector when its signature is visible beyond the influence of the vortex circulation.

In this new paradigm, the GVTD algorithm fits the atmospheric vortex circulation to a new variable VdD/RT in a linear azimuth angle (θ′), rather than the Doppler velocity Vd in a nonlinear angle (ψ), which is used in GBVTD. Key vortex kinematic structures (e.g., mean wind, axisymmetric tangential wind, etc.) in the VdD/RT space simplify the interpretation of the radar signature and eliminate the geometric distortion inherent in the Vd display. This is a significant improvement in diagnosing vortex structures in both operations and research. The advantages of using VdD/RT are illustrated using analytical atmospheric vortices, and the properties are compared with GBVTD. The characteristics of the VdD/RT display of Typhoon Gladys (1994) can be approximated by a constant mean wind plus an axisymmetric vortex.

Corresponding author address: Dr. Wen-Chau Lee, Earth Observing Laboratory, National Center for Atmospheric Research, Boulder, CO 80307-3000. Email: wenchau@ucar.edu

Abstract

The primary circulation of atmospheric vortices, such as tropical cyclones and tornadoes, can be estimated from single-Doppler radar observations using the ground-based velocity track display (GBVTD) algorithm. The GBVTD algorithm has limitations in the following four areas: 1) distortion in the retrieved asymmetric wind fields, 2) a limited analysis domain, 3) the inability to resolve the cross-beam component of the mean wind, and 4) the inability to separate the asymmetric tangential and radial winds. This paper presents the generalized velocity track display (GVTD) algorithm, which eliminates the first two limitations inherent in the GBVTD technique and demonstrates the possibility of subjectively estimating the mean wind vector when its signature is visible beyond the influence of the vortex circulation.

In this new paradigm, the GVTD algorithm fits the atmospheric vortex circulation to a new variable VdD/RT in a linear azimuth angle (θ′), rather than the Doppler velocity Vd in a nonlinear angle (ψ), which is used in GBVTD. Key vortex kinematic structures (e.g., mean wind, axisymmetric tangential wind, etc.) in the VdD/RT space simplify the interpretation of the radar signature and eliminate the geometric distortion inherent in the Vd display. This is a significant improvement in diagnosing vortex structures in both operations and research. The advantages of using VdD/RT are illustrated using analytical atmospheric vortices, and the properties are compared with GBVTD. The characteristics of the VdD/RT display of Typhoon Gladys (1994) can be approximated by a constant mean wind plus an axisymmetric vortex.

Corresponding author address: Dr. Wen-Chau Lee, Earth Observing Laboratory, National Center for Atmospheric Research, Boulder, CO 80307-3000. Email: wenchau@ucar.edu

1. Introduction

Atmospheric vortices such as tropical cyclones and tornadoes possess a dipole Doppler velocity pattern when observed by a ground-based Doppler radar scanning in a plan-position indicator (PPI) mode (e.g., Donaldson 1970). The shape of the dipole Doppler velocity pattern of an axisymmetric vortex is a function of the distance between the “vortex circulation center” (hereafter, the center) and the radar, core diameter, and the ratio of peak tangential to peak radial wind. The dipole rotates clockwise (counterclockwise) when the radial wind is inflow (outflow), as shown in Wood and Brown (1992). When an axisymmetric vortex is located at infinite distance from the radar, its center can be determined as the midpoint of the line segment connecting the two peak dipole velocities (Wood and Brown 1992). As the vortex approaches the radar, the peak velocities of the dipole move toward the radar faster than the center. Hence, the dipole pattern is distorted and the center does not fall on the line segment connecting the two peak velocities of the dipole, which increases the complexity of accurately identifying the center in operational setting.

Based on the rotational characteristics of a vortex, Lee et al. (1999) formulated a single-Doppler wind retrieval methodology, called the ground-based velocity track display (GBVTD), to retrieve the primary kinematic structures of atmospheric vortices (Lee et al. 2000; Roux et al. 2004; Bluestein et al. 2003; Harasti et al. 2004; Lee and Wurman 2005; Lee and Bell 2007; Tanamachi et al. 2007). The symbols and geometry of the GBVTD technique are illustrated in Fig. 1. For the convenience of discussion, we define the term “RT vector” with a magnitude of RT and a direction pointing from the radar toward the center. Using a cylindrical coordinate system with the center as the origin, the GBVTD technique performs a Fourier decomposition of the Doppler velocity Vd around each circle of radius R, and then estimates the three-dimensional (3D) tangential and radial circulations that cannot be deduced by existing single-Doppler wind retrieval methods (e.g., Browning and Wexler 1968; Donaldson 1991; Harasti 2003). Plausible axisymmetric 3D kinematic and dynamic quantities, such as the angular momentum, vertical vorticity, and perturbation pressure, can also be computed from the GBVTD-retrieved axisymmetric tangential and radial winds (Lee et al. 2000; Lee and Wurman 2005; Lee and Bell 2007).

The limitations of the GBVTD technique are as follows: 1) distortion in the retrieved asymmetric wind fields, 2) a limited analysis domain, 3) an inability to resolve the cross-beam component of the mean wind, and 4) an inability to separate the asymmetric tangential and radial winds. The first three limitations are caused by the sampling geometry, while the last is due to the intrinsic closure assumptions of the GBVTD technique. Hence, the GBVTD-derived vortex circulation is a proxy of the “true” circulation and may inherit large uncertainties resulting from the above limitations in certain situations (Lee et al. 1999).

This paper presents the generalized velocity track display (GVTD) technique and its applications to atmospheric vortices. The GVTD technique extends the foundation of GBVTD established in Lee et al. (1999) in an attempt to address the first three aforementioned limitations inherent in the GBVTD technique. Starting from the same radar observations, the GVTD technique introduces a new variable VdD/RT by multiplying the distance of each gate (D) by Vd, and then scaling by the distance between the radar and the vortex center (RT). Key vortex kinematic structures displayed in the VdD/RT space simplify the interpretation of the radar signature and eliminate the geometric distortion inherited in the Vd space (Jou et al. 1996). It will be shown that GVTD expands VdD/RT into Fourier coefficients in a linear coordinate (θ′) rather than expanding Vd in a nonlinear coordinate (ψ′) in GBVTD. This results in a slightly complicated but mathematically exact representation, eliminating the required approximation of cosα in GBVTD [Eq. (5) in Lee et al. (1999)]. GVTD is able to retrieve asymmetric vortex structures without distortion when the center is known accurately. The GVTD formulation can be applied to the extensions of the velocity track display (VTD) techniques (e.g., Roux et al. 2004; Liou et al. 2006) to improve their performance.

Section 2 describes the mathematical formulation of the GVTD technique. In section 3, characteristics of analytical wind patterns displayed in VdD/RT space are given and compared with those displayed in Vd space. Section 4 compares the center and radius of maximum wind (RMW) estimates between the VdD/RT and Vd displays. In section 5, we illustrate and compare the wind fields retrieved from the GVTD and GBVTD techniques of several analytical vortices. Error analyses were conducted to investigate the differences between GVTD- and GBVTD-retrieved wind fields in the presence of asymmetry, a misplaced center, and uncertainty in the mean wind. Section 6 uses a simple flow model to simulate Typhoon Gladys’s observed characteristics. A summary and recommendations for future work are given in the final section.

2. The generalized VTD technique

We begin with the horizontal projection of the Doppler velocity [Eq. (3) in Lee et al. (1999)],
i1520-0493-136-3-995-e1
where ϕ is the elevation angle. Applying the relations shown in Fig. 1, ψ = θ′ − α = θ − θT − α = θ − θd, we obtain
i1520-0493-136-3-995-e2
For a Doppler velocity at point E (D, θd) (Fig. 1), one may deduce that
i1520-0493-136-3-995-e3
i1520-0493-136-3-995-e4
Note that all angles are mathematical angles where positive is defined as being counterclockwise.
Substituting (3) and (4) into (2) and approximating d/cosϕ with Vd, we obtain
i1520-0493-136-3-995-e5
Rearranging (5) and applying trigonometry identities, we obtain
i1520-0493-136-3-995-e6
Let θ′ = θ − θT and use the relation θ − θM = (θ − θT) + (θT − θM), and (6) becomes
i1520-0493-136-3-995-e7

For a given R, the right-hand-side of (7) depends only on θ′. Comparing Eq. (7) in Lee et al. (1999) with (7) herein, it can be seen that Vd, a function of nonlinear ψ in GBVTD, corresponds to VdD/RT, a function of linear θ′ in GVTD. Note that Eq. (7) in Lee et al. (1999) required an approximation to link the unknown variable sinα and the known constant sinαmax = R/RT. When R > RT, αmax is not defined1 (see Fig. 1). Explicitly moving D to the left-hand side as part of the new variable makes (7) mathematically exact and valid for all radii beyond R > RT.

Following Lee et al. (1999), we decompose VdD/RT, VT, and VR into Fourier components in the θ′ coordinates:
i1520-0493-136-3-995-e8
i1520-0493-136-3-995-e9
and
i1520-0493-136-3-995-e10
in which An (VTCn and VRCn) and Bn (VTSn and VRSn) are the azimuthal wavenumber n cosine and sine components of VdD/RT (VT and VR), as defined in Lee et al. (1999).
Substituting (8), (9), and (10) into (7), similar to Lee et al. (1999), we obtain the following:
i1520-0493-136-3-995-e11
i1520-0493-136-3-995-e12
i1520-0493-136-3-995-e13
i1520-0493-136-3-995-e14
i1520-0493-136-3-995-e15
Rearranging (11)(15) to express each wave component of the vortex using these Fourier coefficients, we have
i1520-0493-136-3-995-e16
i1520-0493-136-3-995-e17
i1520-0493-136-3-995-e18
i1520-0493-136-3-995-e19
i1520-0493-136-3-995-e20

Equations (16)(20) correspond to Eqs. (19)–(27) in Lee et al. (1999)2 for GBVTD with additional terms associated with R/RT. In the limit of R/RT ∼ 0, these two sets of equations are identical when they are truncated at the same wavenumber n. It can be shown (appendix A) that (6) reduces to VTD [Eq. (3) in Lee et al. 1994] in the limit of R/RT ∼ 0 (i.e., D/RT ∼ 1). In this situation, all radar beams of ground-based radar can be treated parallel with each other, similar to the sampling geometry in VTD. In addition, the most severe geometric constraint imposed in GBVTD (Lee et al. 1999), that is, the analysis domain of a storm is limited to R/RT < 1, is no longer a constraint. The analysis domain in the GVTD extends over the entire domain wherever sufficient Doppler velocity data are available to yield reliable GVTD Fourier coefficient estimates. This point will be illustrated in section 5. Therefore, GVTD is a more general form of the VTD family of techniques.

GVTD faces similar problems encountered in Lee et al. (1999) where the numbers of unknown variables are greater than the number of equations. Hence, we assume the same closure assumptions as GBVTD, namely, that the asymmetric VR is smaller than VT and therefore can be ignored. Searching for dynamic closure assumptions for the VTD family of techniques remains a research topic.

3. Characteristics of the VdD/RT display

The characteristics of vortex signatures in VdD/RT space can be evaluated analytically from (7). Because (7) is similar to the VTD [Eq. (3) in Lee et al. 1994], characteristics of VdD/RT resemble those of Vd in VTD where radar beams are parallel to each other and there is no geometric distortion of the asymmetric structures.

Regrouping (7) yields
i1520-0493-136-3-995-e21
in which
i1520-0493-136-3-995-e22
i1520-0493-136-3-995-e23
i1520-0493-136-3-995-e24

It can be concluded that for a fixed R, (21) is a function of θ′ only as long as VM, VT, and VR are functions of θ′. The existence of VR and/or VM in (23) raises or lowers the entire sine curve. Note that in VTD and GBVTD, VM is the only factor that would shift the entire curve up and down for axisymmetric vortices. In (24), θ0 represents the phase shift of the sine curve (i.e., azimuthal rotation of the dipole). If there is no mean wind (VM = 0), then θ0 reduces to tan−1(VR/VT), as in GBVTD (Fig. 2 in Lee et al. 1999). The effect of VM on θ0 is further reduced by the factor R/RT in the near-core region, but this effect may not be ignored at far radii of the vortex. If VTVR, then θ0 ∼ 0. Thus, in a vortex without significant VR, θ0 is generally small. Note that the phase shift of the dipole signature does not depend on VM in Vd space, but does depend on VM in VdD/RT space. This has the effect of complicating the estimation of the axisymmetric radial wind, as can be seen by comparing (18) with Eq. (21) of Lee et al. (1999). The mean wind vector can be estimated by using the hurricane volume velocity processing (HVVP) method (Harasti 2003) or using the unique signature of the mean wind in the VdD/RT display (shown below).

Following Brown and Wood (1991), an idealized vortex flow field is constructed to simulate the wind patterns in Vd and VdD/RT. The complete flow fields include a uniform mean wind, an axisymmetric VT, and an axisymmetric VR. The mathematical expressions in natural coordinates are
i1520-0493-136-3-995-e25
in which t is the unit vector in the tangential direction (positive counterclockwise) and r is the unit vector in the radial direction (positive toward center); VTmax (VRmax) is the maximum axisymmetric VT (VR). Figure 2 shows a set of the flow fields, in which a plus sign marks the center at (x, y) = (60 km, 60 km), VTmax= 40 m s−1, VRmax= 10 m s−1, VM =10 m s−1, θM = 180°, and Rmax = 20 km. For a Rankine vortex, we have λt = λr = 1 when RRmax, and λt = λr = −1 when R > Rmax. The hypothetical Doppler radar is located at the origin.

A constant easterly mean wind and its corresponding Vd and VdD/RT displays are illustrated in Figs. 2a(1) –(3). The mean wind signature is a set of straight lines diverging from the radar in the Vd display [Fig. 2a(2)]. The wind direction is perpendicular to the zero Doppler velocity line pointing toward the positive contours and the wind speed is the maximum Doppler velocity in the domain. In the VdD/RT display, the easterly mean wind signature is a set of north–south-oriented parallel lines [Fig. 2a(3)]. It can be shown (appendix B) that the mean wind vector is the gradient of VdD. Note that RT is a scale factor and is not needed to determine the mean wind vector. This parallel line signature can be identified by visually examining the VdD/RT contours not affected by the vortex circulation, usually in the quadrant opposite the center. An objective procedure to deduce the mean wind vector when mixed with a vortex will be presented in the forthcoming second part of this work. Hence, one of the unresolved quantities in the GBVTD formulation, the cross-beam mean wind, can be directly estimated in the VdD/RT display.

The flow fields Vd and VdD/RT displays of an axisymmetric vortex are portrayed in Figs. 2b(1) –(3). The striking differences between the Vd and VdD/RT displays [Figs. 2b(2), (3)] are in the shapes of the contours. The Vd pattern of an axisymmetric tangential vortex [Fig. 2b(2)] is distorted as a function proportional to R/RT. On the contrary, the VdD/RT contours are symmetric about the center [Fig. 2b(3)], independent of R/RT with no distortion. Jou et al. (1996) first proposed using the midpoint of the line connecting the dipole in the VdD/RT display to estimate the center and the RMW [they called it the “velocity distance azimuth display” (“VDAD”) method].

Examples of the axisymmetric radial outflow are illustrated in Figs. 2c(1) –(3). When considering VT = 0, VM = 0 in (25), U1 = VR, U2 = VR(R/RT), and θ0 = π/2 or 3π/2, (21) becomes VdD/RT = −VRsin(θ′ − π/2) + VR(R/RT). This is the reason why the VR signature in the VdD/RT display is not symmetric about the center and there is a π/2 phase difference between VR and VT in VdD/RT displays. Nevertheless, the contours are more symmetric in the VdD/RT display compared with the Vd display.

Figures 2d(1) –(3) illustrate the flow field of a combination of VM, VT, and VR, and the corresponding Vd and VdD/RT displays. The combined flow field is asymmetric. However, the dipole is not significantly distorted in the VdD/RT display near the RMW, even with the addition of a constant VM and axisymmetric VR, allowing the center and RMW to be estimated using the VDAD method [Fig. 2d(3)].

To further examine the characteristics of vortex signatures in the Vd and VdD/RT displays as a function of Rmax, two axisymmetric rotating vortices with VTmax = 50 m s−1 and an Rmax of 30 and 80 km are constructed, and their corresponding Vd and VdD/RT displays are shown in Fig. 3. Figures 3a,c illustrate the Vd and VdD/RT displays of the smaller vortex with an Rmax of 30 km where the radar is located outside the RMW, while Figs. 3b,d portray the corresponding profiles of Vd and VdD/RT around two radii (at R = 30 and 60 km). As R increases, the peak wind locations in the Vd display (ψ = π/2 and 3π/2 in Fig. 3b) shift toward ψ = θ′ = π, while the peak values of VdD/RT (Fig. 3d) remain at θ′ = π/2 and θ′ = 3π/2. In the VdD/RT display, the center remains at the intersection between the zero Doppler velocity line and the line connecting the dipole, independent of the geometric factor R/RT.

When the radar is inside the RMW of the larger vortex (Rmax = 80 km, RT = 70.7 km; hence Rmax > RT), the radar does not sample the full component of the VTmax; therefore, the peak Vd around the RMW (blue circle in Fig. 3e) is less than the VTmax (Fig. 3a). However, the corresponding VdD/RT profile at the R = Rmax = 80 km and R = 110 km (blue line and red line in Figs. 3g,h) can recover the vortex intensity as in the Rmax < RT case (Fig. 3c). The dipole structure can be fully recovered in the VdD/RT space, and even the radar does not sample the full component of VT at each radius. This property can be illustrated analytically by setting VR = 0, VM = 0, U1 = VT, U2 = 0, and θ0 = 0 in (21); we will then have VdD/RT = −VT sinθ′. There is a clear advantage to displaying atmospheric vortices in VdD/RT space over the traditional Vd space.

In summary, representing a vortex in VdD/RT space simplifies the vortex signatures and eliminates the dipole distortion as a function of Rmax/RT in the traditional Vd display. In particular, the signature of a constant mean wind is a set of parallel lines. The potential to separate the vortex and the mean wind in VdD/RT display provides a new paradigm to study the interaction between the vortex and the mean flow.

4. Center and RMW

It can be shown from (21) that the center is the midpoint of the line connecting the dipole in the VdD/RT display (i.e., the VDAD method) as long as U1 and U2 remain constant at the Rmax (i.e., any combination of axisymmetric VT, axisymmetric VR, and a constant VM). The existence of axisymmetric VR and/or VM would add a constant magnitude and a constant phase shift to the sine curve at each radius that makes the dipole uneven in magnitude and rotates in azimuth. It is found that the VDAD method is especially useful for identifying the center of a near-axisymmetric vortex in a real-time operational environment. When significant asymmetric components exist, (21) is not valid and accurately estimating the center will require a more elaborate methodology, such as the “simplex” method (Lee and Marks 2000), which is beyond the scope of this paper.

We applied the Wood and Brown (1992) method to retrieve the center and Rmax [Fig. 2d(2)], where the estimated center is located at (60.33 km, 60.19 km) and Rmax is 19.65 km, compared with the true center located at (60 km, 60 km) and an Rmax of 20 km. These errors are quite small (the center error is 0.38 km and the Rmax error is 0.35 km). Next, we consider a more extreme case, for example, Rmax increases to 30 km, VTmax decreases to 25 m s−1, VM increases to 20 m s−1, and the direction of VM is from the southwest, parallel to the RT vector. Then, the retrieved center is (61.45 km, 60.34 km) and Rmax is 28.77 km. The errors increase to 1.49 and 1.23 km for the center and RMW, respectively. It is clear that the errors depend both on the assigned wind fields and the relative magnitude of the mean wind speed and direction. On the contrary, both centers estimated using the VDAD method are nearly perfect.

5. GVTD and GBVTD

A series of numerical experiments (using analytical vortices) were conducted to investigate the differences between GVTD- and GBVTD-retrieved wind fields in the presence of 1) asymmetry, 2) a misplaced center, and 3) uncertainty in the mean wind. The design of these experiments is listed in Table 1.

a. The asymmetry test (AS series)

In the asymmetry sensitivity test (AS series), the experimental design follows Lee et al. (1999), where the basic axisymmetric vortex is constructed as follows:
i1520-0493-136-3-995-eq1
where VTmax=50 m s−1, Rmax = 30 km, δ1 = 0.1 s−1, and δ2 = 3 m s−1, respectively.
Four experiments were conducted, including the axisymmetric vortex (AS0), and wavenumber one, two, and three asymmetries (AS1, AS2, and AS3) embedded within the axisymmetric vortex. The asymmetric structures (wavenumbers n = 1, 2, and 3) were constructed using the following equations and the parameters listed in Table 1:
i1520-0493-136-3-995-eq2
where An = 0.2. Note that we still assume that there is no asymmetric radial component in the simulated vortex following Lee et al. (1999).

Figure 4 shows the analytic, GBVTD-, and GVTD-retrieved wind fields for wavenumber zero (first row), wavenumber zero plus one (second row), wavenumber zero plus two (third row), and wavenumber zero plus three (fourth row) asymmetries. The pronounced distortions of the GBVTD-retrieved asymmetric winds [middle column, as discussed in Lee et al. (1999)] are nearly nonexistent in the GVTD-retrieved asymmetric winds (right column), especially in the wavenumber two and three asymmetries. The white-colored area (beyond R = 75 km) in the middle column is not due to zero GBVTD-retrieved winds but rather to no GBVTD estimates, because of the R/RT < 1 restriction in GBVTD; thus, the advantages of GVTD over GBVTD are clearly illustrated.

Figure 5 shows the percentage error distribution of the retrieved wind as a function of θ′ (x axis) and R (y axis) from wavenumber two and three asymmetries only (the errors in wavenumber zero and one cases are negligible, not shown). For GBVTD (Figs. 5a,b), the errors are positively correlated with R and are also highly dependent on the phase of the asymmetry. In general, the wavenumber two vortex was retrieved quite well by the GBVTD technique [Fig. 4c(2)]. The worst errors (>10%) occur along the RT vector beyond R = 40 km. For wavenumber three (Fig. 5b), GBVTD could not retrieve the peak wind along the RT vector beyond the center [θ = 0, see Fig. 4d(2)], while significant phase and amplitude distortions occurred at large R.

In contrast, these radius- and phase-dependent error distributions are not found in GVTD (Figs. 5c,d) and the errors are negligible (<1%), except for regions near the center. Hence, the GVTD analysis is quite robust and nearly eliminates the geometric distortions in the retrieved asymmetric wind fields.

b. The center displacement test (C series)

Figures 6a–c show the GBVTD- and GVTD-retrieved vortex structures when the center is displaced (a) 1, (b) 5, and (c) 10 km, along the RT vector (y axis) away from the true center. The original vortex contains only axisymmetric VT. It can be seen that both algorithms generate apparent wavenumber one components that occur in the opposite direction to the center displacement in all cases, while the amplitude increases as the center displacement increases. These errors are analogous to aliasing errors in signal processing. For a 1-km center displacement, the error is small (not shown). Figure 7 shows that for 5- and 10-km center displacements, however, the errors near the RMW in the GBVTD- and GVTD-retrieved wind fields can be as large as 50% of the analytic axisymmetric vortex (Figs. 7c,d). These results strongly suggest that both methods are sensitive to the center uncertainties, but with similar error characteristics. To have a reasonably correct vortex wind retrieval (e.g., less than 20% of its axisymmetric tangential component), the uncertainty in the center cannot exceed 5 km.

To examine further, we calculated the root-mean-square error (RMSE) of the GBVTD- and GVTD-retrieved VTmax and VRmax for various center displacements. It can be seen that the RMSE of the retrieved VTmax as a function of center displacement in the x and y directions (Figs. 8a,b) is quasi-linearly proportional to the magnitude of the misplaced centers. The error in the GVTD-retrieved VTmax is about 40% less than the GBVTD-retrieved VTmax. A 2-km center displacement produces about a 3% error (1.5 m s−1 error for VTmax = 50 m s−1) in GBVTD and a 2% error in GVTD. The errors are symmetric when the center is displaced perpendicular to the RT vector. When the center is misplaced along the RT vector, the errors are larger (smaller) around the near (far) side of the center. The error distributions of VRmax are very different between the two methods (Figs. 8c,d). The VRmax errors in GBVTD are more symmetric to the center while the VRmax errors in GVTD are more sensitive to the center displacement perpendicular to the RT vector. In a typical situation where the misplaced center is ∼2 km (Lee and Marks 2000), both methods perform very well.

c. The mean wind sensitivity (VM series)

The sensitivity of the GVTD-retrieved axisymmetric vortex on the uncertainty of the mean wind in the direction perpendicular to the RT vector is illustrated in Fig. 9. The error distributions are quite different between the retrieved VTmax and VRmax. It is clear that the retrieved VTmax is sensitive to the error in the mean wind speed. A 50% error in the mean wind speed results in ∼10% error in the retrieved VTmax. The error of VTmax increases proportionally as the assigned error in the mean wind speed. However, the error of VRmax is more sensitive to the mean wind direction instead. The situation is reversed while the mean wind direction is along the RT vector (VM2 test, not shown); the retrieved VTmax is more sensitive to the mean wind direction and the retrieved VRmax is more sensitive to the mean wind speed.

6. Typhoon Gladys

In this section, we use Typhoon Gladys (1994) to gain understanding of the mean wind and vortex signatures in the VdD/RT display. According to the Joint Typhoon Warning Center (JTWC), Gladys was a relatively small typhoon with moderate intensity. The Vd constant-altitude PPI (CAPPI) display of Gladys at 4-km height (Fig. 10a) shows that Gladys’ inner-core diameter is about 35 km, indicated by the circle in the lower-right-hand corner of the display. The approaching Doppler velocity exceeded 50 m s−1 and the receding component was about 15 m s−1. This pronounced asymmetric structure indicates a possible combination of a strong mean flow and/or an asymmetric vortex. Figure 10b shows the corresponding VdD/RT display. It is clear that the vortex circulation was mostly confined to lower-right corner of the display, where the near-parallel straight lines aligned in a north–south direction to the left of the radar (opposite side of the center) suggested a likely east–west-oriented mean wind at this level.

The flow field of a Gladys-sized Rankine vortex with a RMW of 16.5 km and VTmax of 35 m s−1 embedded in a 20 m s−1 easterly mean wind is simulated, and the corresponding Vd and VdD/RT displays are shown in Figs. 10c,d. Even with no asymmetric VT and VR in the simulation, the similarity between the observed and simulated Vd (Figs. 10a,c) and VdD/RT (Figs. 10b,d) is very encouraging. With the VdD/RT display, the gross features of the vortex and its accompanied mean flow characteristics can be estimated with a reasonable accuracy, while the mean wind is not straightforward enough for identification in the Vd display (Fig. 10a). Note that an east–west-oriented convective line ∼70 km north of the radar forces the VdD/RT contours to be oriented in the east–west direction in Fig. 10b instead of north–south, as in Fig. 10d. Differences in the actual and simulated VdD/RT are also apparent in the rainbands northeast of the radar where asymmetric vortex components are likely.

7. Summary and future work

This paper introduces the GVTD technique with a new variable VdD/RT as the new paradigm to display, interpret, and retrieve kinematic structures of atmospheric vortices. Using analytical vortices, the properties of the VdD/RT display of atmospheric vortices and the GVTD technique are examined and compared with the radar signatures in the Vd display and the GBVTD technique. It is evident that the VdD/RT display simplifies the vortex interpretation and eliminates the geometric distortion of the dipole signature displayed in Vd. It is shown that GVTD is a more general form for the VTD family of techniques. The advantages of the GVTD technique over the GBVTD technique are as follows:

  1. Negligible geometric distortion: The VdD/RT variable relates the vortex circulation in a linear coordinate system. Hence, the pronounced distortion of retrieved asymmetric winds in GBVTD has been nearly eliminated, especially when high-wavenumber asymmetries are involved and/or R/RT ∼ 1.
  2. Expanded analysis domain: In GBVTD, the analysis domain is limited by R/RT < 1, where the distortion of the retrieved wind fields worsens as R/RT approaches unity. In GVTD, the analysis can be extended to cover the entire domain of the Doppler radar whenever there are enough data for meaningful GVTD analysis, as portrayed in Fig. 4. The ability to recover the dipole structure for R > RT is particularly striking. This characteristic is especially important for assimilating GVTD-retrieved winds into a numerical model in the future.
  3. Relatively straightforward: The subjective estimation of the mean wind is from the VdD/RT display when the vortex circulation is not dominating the Doppler velocities. In this situation, a constant mean wind appears as parallel lines and can be easily recognized subjectively. The possibility to separate the vortex signature from the mean wind signature provides a useful tool for studying the vortex mean flow interactions in the future.

When estimating the center location and RMW in the VdD/RT space, the VDAD method has advantages over the Wood and Brown (1992) method in the Vd space, especially for a near-axisymmetric vortex. The VDAD method is particularly useful in an operational environment for quick determination of the gross features of the vortex. A more quantitative and robust algorithm to estimate the center location and RMW is to combine GVTD and a simplex method that will be presented in a future paper.

Beyond all of these advantages, there are limitations in the GVTD technique worth noting. First, from the center displacement tests (C series), it can be seen that the retrieved VRmax from GVTD is 3 times more sensitive to the accuracy of the center than VRmax derived from the GBVTD technique (Figs. 8c,d). However, in a typical uncertainty of ∼2 km around the center, the errors are not significant. Second, the subjective determination of the mean wind vector will have difficulty when the vortex circulation is large [e.g., Hurricane Katrina (2005)]. Third, the distance weighting of VdD/RT rescales Doppler velocities. It is possible that the missing data at a large range resulting from noise or limited unambiguous range may affect the least squares fit of the GVTD coefficients differently than in the GBVTD technique.

Several research activities are currently underway to evaluate and address these limitations. Taking advantage of the constant mean wind signature in the VdD/RT display, an automated method has been developed to estimate the best mean wind vector and center location simultaneously. We will apply GVTD to real atmospheric vortices to examine the limitations related to real observations (such as missing data, distance weighting, etc.). These topics will be reported in future papers.

Acknowledgments

The authors thank two anonymous reviewers for their careful reading, many constructive comments, and suggestions that drastically improved the quality and clarity of this manuscript. They also thank Dr. H.-M. Hsu, Mr. M. Bell, and Mr. S. Ellis for their helpful discussion. This research is partially supported by the National Science Council of Taiwan, Republic of China, under Grant NSC95-2111-M-002-18-Ap2 and NSC95-2625-Z-002-004. The first author expresses his appreciation for the hospitality of Earth Observing Laboratory of the National Center for Atmospheric Research for hosting his sabbatical leave in the summer of 2004 and the spring of 2005.

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APPENDIX A

VTD and GVTD

This appendix intends to show GVTD is a general form of VTD by proving that (6) is identical to VTD [Eq. (3) in Lee et al. 1994] in the limit of R/RT ∼ 0. Note that the geometry and symbols between airborne Doppler radar (e.g., Fig. 3 in Lee et al. 1994) and ground-based radar in this paper (Fig. 1) are different. Assuming the flight track in VTD is oriented in the east–west direction (Fig. 3 in Lee et al. 1994), it is equivalent to have a θT = π/2 in the GVTD geometry (Fig. 1). The azimuth angle θ in VTD (θVTD) (ϕ in Lee et al. 1994) and θ in GVTD (θGVTD) result in θVTD = θGVTDπ. In addition, positive Vd in VTD (VVTDd) corresponds to positive VT (cyclonic rotation) and VR (radial outflow) of a vortex, while the opposite is true for Vd in GVTD (VGVTDd). In the limit of R/RT ∼ 0, replacing θ with θGVTD and θTθM with π/2 − θGVTDM, (6) becomes
i1520-0493-136-3-995-ea1
which proves GVTD reduces to VTD in the limit of R/RT ∼ 0.

APPENDIX B

Derivation of the Mean Wind Vector

Starting from (5), moving D to the left-hand side, and considering a uniform mean wind only, we have
i1520-0493-136-3-995-eb1
where xT = RT cosθT, yT = RT sinθT, x′ = R cosθ, and y′ = R sinθ (Fig. 1). The origin of the Cartesian coordinate (x, y) is located at the radar. This equation is in the form of a straight line, ax + by = c, because VM and θT are constant for a uniform mean wind.
By taking the gradient of (B1), we have
i1520-0493-136-3-995-eb2
Therefore, the direction of the gradient vector is θM while the magnitude of the gradient vector is VM.

Note that (B2) is independent of RT and is expressed in a Cartesian coordinate system. As a result, estimating the mean wind vector using VdD can be applied to any flow field, and is not limited to atmospheric vortices.

Fig. 1.
Fig. 1.

The geometry and symbols used in the formulation of the GBVTD: T is the vortex center; E is the observation point; Vd/cosϕ is the horizontal projection of the observed Doppler velocity; and VM, VT, and VR are the mean wind, tangential wind, and radial wind of the vortex, respectively (adapted from Lee et al. 1999).

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 2.
Fig. 2.

(left) [a(1)] A constant easterly mean wind with a magnitude of 10 m s−1, [b(1)] a Rankine-combined vortex, [c(1)] the axisymmetric radial wind, and [d(1)] the total wind [(a1) + (a2) + (a3)], and the corresponding (middle) observed Doppler velocity and (right) VdD/RT display of the simulated wind fields in the left-hand side. The Doppler radar is located at the lower-left corner.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 3.
Fig. 3.

A comparison of Vd and VdD/RT displays for two different radii (delineated as red and blue lines) for two vortices with different RMWs: (a) Vd display for a pure rotating vortex with Rmax = 30 km; (b) Vd profiles at R = 30 and 60 km; (c) same as (a), but for VdD/RT display; and (d) same as (b), but for VdD/RT profiles. (e)–(h) Same as (a)–(d), but for Rmax = 80 km > RT and the two Vd profiles are at R = 80 and 110 km. The center “T” is located at (x, y) = (200 km, 200 km) and the hypothetical Doppler radar “O” is located at (x, y) = (150 km, 150 km), with RT = 50√2 km.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 4.
Fig. 4.

Comparison of GBVTD- and GVTD-retrieved vortex structure. [a(1)] The simulated axisymmetric wind field, [a(2)] the GBVTD-retrieved wind field, and [a(3)] the GVTD-retrieved wind field. Same as [a(1)]–[a(3)], but for [b(1)]–[b(3)] wavenumber one, [c(1)–[c(3)] wavenumber two, and [d(1)–[d(3)] wavenumber three cases. The center is located at (0, 0), while the hypothetical Doppler radar is located at (0, −80).

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 5.
Fig. 5.

Percentage error distribution of the retrieved vortex as a function of θ′ (x axis, °) and R (y axis, km) for (a) wavenumber two GBVTD-retrieved VT, (b) wavenumber three GBVTD-retrieved VT, (c) wavenumber two GVTD-retrieved VT, and (d) wavenumber three GVTD-retrieved VT.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 6.
Fig. 6.

Comparison of GBVTD- and GVTD-retrieved pure rotational vortex structure with prescribed center displacement along the RT vector (y axis). Results are for the case of [a(1)] GBVTD- and [a(2)] GVTD-retrieved structure for a center displacement of 1 km. Same as [a(1)] and [a(2)], but for a center displacement of [b(1)], [b(2)] 5 and [c(1)], [c(2)] 10 km, respectively.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 7.
Fig. 7.

Percentage error distribution of the retrieved vortex as a function of θ′ and R. Results are for the case of (a) GBVTD- and (b) GVTD-retrieved vortex with a center displacement of 5 km; (c), (d) same as (a), (b), but for a center displacement of 10 km.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 8.
Fig. 8.

RMSE distributions of the retrieved VTmax and VRmax from GBVTD and GVTD corresponding to center misplacement as a function of x and y from the true center: VTmax retrieved by (a) GBVTD and (b) GVTD, and VRmax retrieved by (c) GBVTD and (d) GVTD.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 9.
Fig. 9.

Effect of mean wind on the vortex retrieved by GVTD. RMSE of (a) retrieved VTmax and (b) retrieved VRmax. The mean wind is perpendicular to the RT vector with a magnitude of 10 m s−1.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Fig. 10.
Fig. 10.

The (a) Vd and (b) VdD/RT CAPPI displays of Typhoon Gladys at 4-km altitude observed by the Civil Aeronautical Administration Doppler radar at northern Taiwan. Simulated (c) Vd and (d) VdD/RT derived from the parameters VM = 20 m s−1, θM = −90° (easterly), VTmax = 35 m s−1, and VR = 0.

Citation: Monthly Weather Review 136, 3; 10.1175/2007MWR2116.1

Table 1.

Summary of sensitivity tests on GBVTD and GVTD. Three test series were conducted to quantify the asymmetry (AS), center displacement (C), and mean wind sensitivities (VM). The prefix G (M) represents results from GBVTD (GVTD). CxN (CyN) represents the response to a misplaced center toward east (north) for N km.

Table 1.

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

1

This restriction does not really exist if R/RT, instead of sinαmax, is used in Eq. (B3) in Lee et al. (1999). However, it can be shown that when R/RT > 1, ψ spans an insufficient and highly nonlinearly spaced subset of 0–2π for a meaningful GBVTD fit.

2

There is a typographical error in Eq. (20) of Lee et al. (1999). The correct equation is VTC0 = −B1B3VM sin(θTθM) sinαmax + VRS2.

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