## Info

Figure 9. (a) Sea surface albedo for radiation between 0.2 and 4.0 /m as a function of the wind speed and the cosine of the solar zenith angle for the clear sky condition with chlorophyll concentrations of 0 (left panel) and 12 (right panel) mg/m3. (b) Transmission at the ocean depth of 5 m with chlorophyll concentrations of 0 (left panel) and 2mg/m3 (right panel) (after Lee and Liou, 2007).

Figure 9. (a) Sea surface albedo for radiation between 0.2 and 4.0 /m as a function of the wind speed and the cosine of the solar zenith angle for the clear sky condition with chlorophyll concentrations of 0 (left panel) and 12 (right panel) mg/m3. (b) Transmission at the ocean depth of 5 m with chlorophyll concentrations of 0 (left panel) and 2mg/m3 (right panel) (after Lee and Liou, 2007).

### 5. Summary

In this paper, we have presented three contemporary and largely unsolved problems in atmospheric radiative transfer with application to the Asian-Pacific region. Although it has been written in a review format, some new results have also been presented to illustrate important points in climate and remote sensing applications.

Aerosols are undoubtedly the most uncertain climate element in the Earth's atmosphere at the present time. Analysis and understanding of the radiative forcings owing to the direct and indirect aerosol climate effects require the basic scattering and absorption properties. The dust storms originating in China during the springtime and their global transport have a profound impact on the climate of East Asia. Moreover, China is the largest underdeveloped country in the world and has been obtaining 80% of its energy from coal combustion. The emission of BC has been especially large owing to low-temperature household coal burning. Dust and BC are nonspherical and inhomogeneous particles. Reliable and efficient determination of the scattering and absorption properties of these particles based on theoretical approach, numerical solution, and controlled laboratory experiment must be undertaken in order to reduce uncertainties in their radiative forcings by means of global and regional climate models.

The effect of terrain inhomogeneity on the radiation fields of the surface and the atmosphere above has not been accounted for in weather and climate models at this point. On the basis of a 3D radiative transfer program we illustrated that without consideration of topographical variability in typical midlatitute mountainous regions in the United States, net solar flux at the surface can be off by 520 W/m2, assuming a flat surface. Many landscapes on the continents exhibit significant terrain features and in particular, the Tibetan Plateau has a profound influence on the general circulation of the atmosphere and climate in the East Asia region. The temporal and spatial distributions of surface radiation over this intensive topography determine surface dynamic processes and atmospheric radiative heating. Thus, accurate calculations and parametrizations of radiative transfer including both the solar and the thermal infrared spectra in clear and cloudy conditions must be developed for incorporation in regional and global models.

The last unsolved radiative transfer area we identified is associated with the transfer of solar radiation in the atmosphere-ocean system. The ocean covers about 70% of the Earth's surface, and the radiation interface between the atmosphere and the ocean is rather intricate. We pointed out that the vertical distribution of solar flux influences the stability and stratification of the mixed layer and the sea surface temperature. The surface roughness produced by the winds is the predominating factor controlling the penetration of solar flux into the ocean mixed layer and it must be correctly modeled for radiation calculations. Finally, we pointed out the lack of fundamental scattering and absorption information on irregular phytoplankton and other species in the ocean for heating rate calculation and remote sensing application.

### Acknowledgements

The research for this work has been supported in part by NSF Grants ATM-0331550 and ATM-0437349 and DOE Grant DE-FG03-00ER62904. We thank Yoshihide Takano and Rick Hansell for their assistance during the preparation of this article.

[Received 4 January 2007; Revised 28 April 2007; Accepted 30 April 2007.]

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Progress in Atmospheric Vortex Structures Deduced from Single Doppler Radar Observations

Wen-Chau Lee

Earth Observing Laboratory, National Center for Atmospheric Research*, Boulder, Colorado, USA [email protected] edu

Ben Jong-Dao Jou

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

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

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

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

### 1. Introduction

Atmospheric vortices, such as tornados, meso-cyclones, and tropical cyclones (TCs), are highly correlated with severe weather and disasters threatening human lives and property. Famous examples are Typhoon Herb (1996), Hurricane Andrew (1992), Hurricane Katrina (2005), and the supertornado outbreak on 3-4 April 1974 in the United States, where hundreds of lives and billions of dollar's worth of property were lost because of damaging winds, mudslides, and floods. The spatial and temporal scales of these vortices span five orders of magnitude (from tens of meters to hundreds of kilometers, and from several minutes to ten days).

Sampling and deducing the two-dimensional and three-dimensional internal structures of these wide ranges of atmospheric vortices have been ongoing challenges for operational forecasts, scientific research, and numerical model initialization and validation (e.g. Houze et al., 2006; Wang and Wu, 2004).

Atmospheric vortex structures have been observed by and inferred from in situ measurements, remote sensing, and photogrammetry (e.g. Davies-Jones et al., 2001; Marks, 2003; Gray, 2003). Doppler radar remains the only instrument that can probe 3D internal structures of cloud and precipitation systems at a spatial resolution of 0.025-1 km and a temporal resolution of 1-30 minutes. Since Doppler radars sample only one component of the 3D motion along a radar beam, estimating 3D wind vectors requires two or more Doppler radars viewing the area of interest simultaneously, in conjunction with the mass continuity equation (e.g. Armijo, 1969). A comprehensive review of the multiple Doppler radar analysis and the subsequent dynamic retrieval techniques can be found in the literature (e.g. Ray, 1990).

Although dual Doppler analyses have provided critical insights into understanding structures of large atmospheric vortices such as TCs, dual Doppler radar analysis remains primarily an interactive and labor-intensive process with limited value for real-time applications. Only recently, encouraging real-time dual Doppler analysis of Atlantic hurricanes was performed on board the US National Oceanic and Atmospheric Administration (NOAA) WP-3D (Gamache et al., 2004). This approach has potential, as the enhanced computer power enables data quality control and allows navigation corrections to be completed in real time. Until then, practical limitations prevent much wider applications to atmospheric vortices, especially in real-time settings.

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

The difficulties in sampling these storms by multiple radars severely limit the monitoring and study of severe weather events associated with tornados, mesocyclones, and TCs over most of the world. As a result, 2D and 3D wind fields of atmospheric vortices can only be estimated from single Doppler radar wind retrieval (SDWR) algorithms. Although the kinematic structures of atmospheric vortices cannot be retrieved from SDWR as accurately and completely as from dual Doppler radar analysis, recent advances in SDWR algorithms have permitted the deduction of primary and secondary vortex structures in real time, expanding the analyses of atmospheric vortices far beyond the restricted dual Doppler radar domain.

Comprehensive reviews on the subject of TCs and severe weather (tornados and meso-cyclones) from the radar perspective have been given by Marks (2003) and Bluestein and Wakimoto (2003). This review article focuses on the progress of SDWR algorithms, specifically their applications, in objectively identifying the vortex circulation center and deducing 2D and 3D structures of atmospheric vortices. Section 2 reviews the single Doppler radar vortex signatures and their interpretation. Section 3 reviews SDWR algorithms that can deduce structures of atmospheric vortices. Section 4 discusses objective vortex center-finding algorithms. The structures of atmospheric vortices revealed from these SDWR algorithms are reviewed in Sec. 5. Section 6 outlines how the wind field retrieved from the VTD algorithm can be used with data assimilation techniques to refine the divergent wind estimate. Section 7 discusses the generalized VTD technique. Section 8 offers outlooks.

2. Interpretation of Single Doppler Radar Vortex Signatures

The single Doppler signature of a vortex is a dipole that has been recognized since the 1960s (e.g. Donaldson, 1970). Traditionally, wind fields of atmospheric vortices have been assumed axisymmetric and modeled by a Rankine combined vortex (e.g. Rankine, 1901), characterized by two regimes: an inner core region with the velocity increasing linearly with the radius (constant vorticity), and an outer potential flow region where the velocity is inversely proportional to the radius (zero vorticity) (Fig. 1).

Figure 1. Top: Single Doppler velocity signature in a plan view of a stationary Rankine combined vortex, where the line XY is at a constant range from the radar. Bottom,: Tangential velocity profile along the axis XY. Assuming a Doppler radar positioned below the bottom of the page, positive velocities are away from the radar, and negative toward (From Lemon et al., 1978.)

Figure 1. Top: Single Doppler velocity signature in a plan view of a stationary Rankine combined vortex, where the line XY is at a constant range from the radar. Bottom,: Tangential velocity profile along the axis XY. Assuming a Doppler radar positioned below the bottom of the page, positive velocities are away from the radar, and negative toward (From Lemon et al., 1978.)

Starting from a Rankine combined vortex, Brown and Wood (1991) demonstrated that Doppler velocity patterns or a ground-based radar is modulated by (1) varying the vortex intensity and the core size, (2) adding a background uniform wind field (simulating an environmental flow), (3) superimposing diverging winds onto the Rankine combined vortex, and (4) simulating double vortices and divergence sources. Figure 2 illustrates the signatures of a Rankine combined vortex and an axisym-metric radial flow as a function of the distance between the radar and the vortex, where the dipole pattern is distorted as the vortex moves closer to the radar (Wood and Brown, 1992; Lee et al., 1999). The intensity and core diameter of the vortex can be estimated as the magnitude and size of the dipole. Wood and Brown (1992) showed geometrically and mathematically that (1) the dipole rotated counterclockwise when

Figure 2. Horizontal (a) vortex and (b) divergence flow fields. The wind vector length is proportional to wind speed. The true circulation center is indicated by the small dot at the center of each panel. Variations of axisymmetric (a) vortex and (b) divergence Doppler velocity signatures as a function of distance north of the radar are shown beneath the respective flow fields. Solid curves represent flow away from the radar (positive Doppler velocities), short dashed curves represent flow toward the radar (negative Doppler velocities), and the thick, long-dashed curve represents flow normal to the radar viewing direction (zero Doppler velocities). Doppler velocity values are normalized relative to the peak wind speed value with contour intervals of 0.2. Radar is located at 100, 4, and 2 core radii south of the flow field (aspect ratios of 0.02, 0.5, and 1.0, respectively). Distances in the x and y directions have been normalized by the core radius. In the last two panels, radar location is indicated by the large dot. (From Brown and Wood, 1991.)

a diverging wind was superimposed onto a Rankine combined vortex, (2) the Doppler velocity patterns deformed when the aspect ratio, defined as the ratio of the vortex core diameter and the distance between the radar and the vortex center, increases (Fig. 2), and (3) the true vortex center and core diameter can be derived from the distorted Doppler velocity pattern with a correction factor.

The dipole signature has been used to identify atmospheric vortices of different scales and interpret their structures, including the mesocyclones associated with a rotating updraft core in supercell thunderstorms (e.g. Burgess, 1976), mesocyclones and tornados (e.g. Brown et al., 1978; Lemon et al., 1978), mesoscale vortices within mesoscale convective systems (e.g. Stirling and Wakimoto, 1989; Houze et al., 1989), and TCs (e.g. Baynton 1979; Wood and Marks, 1989). The Doppler velocity signatures in real atmospheric vortices in these studies [e.g. Figs. 3(a) and 3(b)] apparently deviate from that of the Rankine combined vortex illustrated in Fig. 2, including: asymmetric dipole magnitude, the zero Doppler velocity isodop which does not bisect the velocity dipole, the curved and nonparallel isodop inside the core region, and azimuthal rotation of the dipole. Attempts have been made to recognize these deviations from the Rankine combined vortex radar signatures so as to infer vortex structure (e.g. Brown and Wood, 1991; Wood and Brown, 1992; Lee et al., 1999).

Desrochers and Harris (1996) proposed a more sophisticated elliptical flow model including both the rotational and divergent winds to simulate the elongated Doppler velocity patterns of the Del City, 20 May 1997, OK mesocyclone. Lee et al., (1999) simulated Doppler velocity patterns of a Rankine vortex plus a mean flow, an axisymmetric radial flow, and tangential asymmetries from wave number 1 to 3. More importantly, the Doppler velocity pattern of an asymmetric vortex varies as a function of the radar's viewing angle

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

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

(Desrochers and Harris, 1996) and/or the phase of the asymmetric component of the vortex (Lee et al, 1999).

In summary, these Doppler velocity patterns were quite similar, and thus it is difficult to distinguish their differences visually. Although qualitative vortex structures can be inferred directly from the shape and magnitudes of the dipole signatures, deducing quantitative vortex structures is not straightforward, and will rely on more sophisticated SDWR algorithms.

### 3. SDWR Algorithms

Retrieval of the 2D and 3D wind fields from single Doppler observations commonly follows one of two approaches: fitting the observed radial velocities to some simplified wind model, or supplementing the radar data with some physics (e.g. conservation, "pseudo conservation," momentum or vorticity equations) and seeking a flow field that satisfies the imposed constraints (Boccippio, 1995).

The first approach typically fits the measured radial velocities to a simple and specific (linear, nonlinear, or circular) model of the wind fields that are representative of a weather phenomenon. In the radar community, the well-known approaches are the velocity azimuth display (VAD) and volume velocity-processing (VVP) methods, where the analysis is radar-centered and assumes a linearly varying horizontal wind field in the radar sampling domain (e.g. Browning and Wexler, 1968; Caya and Zawadzki, 1992; Donaldson, 1991; Matejka and Srivastava, 1991; Waldteufel and Corbin, 1979; Koscielny et al., 1982; Boccippio, 1995). A related approach is the velocity track display (VTD) method, which fits the Doppler velocity measurements to a circular model of the wind field centered at the vortex (Lee et al., 1994; Jou et al., 1994; Roux and Marks, 1996; Lee et al., 1999; Roux et al., 2004; Liou et al., 2006).

The second approach usually constrains the single Doppler velocity field with physical equations, numerical models, or conservation of reflectivity. These SDWR algorithms typically involve multiple volumes of radar data at consecutive times. Assuming Lagrangian conservation of reflectivity, the tracking radar echoes by correlation (TREC) algorithm estimates the movement of the reflectivity field by finding the maximum cross-correlation of features in sequential radar scans or tracking echo centroids (e.g. Zawadzki, 1973; Rinehart, 1979; Symthe and Zrnic, 1983; Tuttle and Foote, 1990; Gall and Tuttle, 1999). More complicated algorithms involve a combination of the Lagrangian conservation of reflectivity and some simple assumptions about the behavior of the radial velocity field (e.g. Qiu and Xu, 1992; Xu et al, 1994; Shapiro et al., 1995; Liou, 1999a,b). The solution techniques are thus based on simple prognostic equations rather than the full set of governing equations to achieve lower computation cost and faster processing time. Recently, Lee et al. (2003) proposed the use of the mesoscale vor-ticity method (MVM) to deduce the 3D TC structures, including divergent wind and vertical velocity in the inner core region, from a frequently observed vorticity field derived from the VTD technique.

The general impression is that favorable agreement between the wind fields retrieved from these SDWR algorithms and independent observations can be achieved when the characteristics of the wind fields reasonably match the underlying geometrical and/or physical assumptions. These techniques tend not to capture the fine-scale features of the flow (i.e. higher order and/or nonlinear features), and the uncertainty of the circulation in the cross-beam (unmeasured) direction is worse than in the along-beam (measured) direction. The wind fields retrieved from SDWR are only as good as the assumptions used in each specific situation. In general, the algorithms in the second approach are capable of resolving more complex wind fields than the algorithms in the first approach. However, when conservation of reflectivity (a commonly used assumption in the second approach) breaks down, such as in the inner core region of atmospheric vortices, the algorithms in the first approach hold the advantage. Only those SDWR algorithms relevant to atmospheric vortices will be reviewed in this section.

### 3.1. The TREC technique

The TREC technique was first proposed by Rinehart and Garvey (1978), with modifications and improvements by Tuttle and Foote (1990). TREC works by storing two scans of low-elevation angle plan position indicator (PPI) or constant altitude PPI (CAPPI) reflectivity data measured at the same elevation angle (or at constant altitude) at two different times (Fig. 4). The time difference between two volumes is typically a few minutes for operational radars. The domain at "Time 1" is divided into equal-sized 2D arrays spaced some distance apart. Each initial array is correlated with all possible arrays of the same size at "Time 2" to find the best-matching second array with the highest correlation coefficient. The wind vector of that point is determined by the distance between the first and the second array divided by the time differences between these two scans. Such techniques are well suited for clear air returns and con-vective storms that can produce wind estimates,

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

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

even when the Doppler velocities are not reliable or not available (for example, beyond the Doppler range) — a significant advantage over Doppler-velocity-based SDWR algorithms. Within the Doppler range, TREC-derived winds can be combined with Doppler velocity data to improve its quality (Gall and Tuttle, 1999). When TREC-derived winds are computed at multiple altitudes, 3D kinematic structures can be obtained.

Gall and Tuttle (1999) applied TREC to three TCs — Hurricane Hugo (1989), Hurricane Erin (1995), and Typhoon Herb (1996) — and obtained reasonable cyclonic circulations (e.g. Typhoon Herb in Fig. 5). TREC tends to underestimate the wind speed in the eyewall compared with the corresponding single Doppler velocities, especially in well-organized, intense eyewalls such as Hugo and Herb. The authors attributed this discrepancy to the following factors: (1) the motion of the radar echo (precipitation features) in the eyewall may be

governed by low-level convergence, and not by the wind (i.e. violation of the intrinsic assumptions of TREC); (2) in a vortex with high curvature in the wind field, it may be better for TREC to be computed in cylindrical coordinates rather than in Cartesian coordinates; (3) relatively uniform reflectivity structures in the azi-muthal direction make it difficult for TREC to find unique features to track; and (4) strong radial shear may exist across the analysis box (15km square in their examples). The first and third factors may be the most difficult to overcome in obtaining reasonable winds in the critical eyewall region. Nevertheless, TREC is effective in deducing the wind fields away from the eyewall of a TC, especially in determining the 17ms-1 wind radii.

### 3.2. The VTD family of techniques

The term VTD (velocity track display), coined by Carbone and Marks (1988), refers to a display of the NOAA WP-3D tail-mounted Doppler radar velocities at the flight level during a radial hurricane penetration through its center [e.g. Hurricane Gloria (1985), Fig. 6]. When a TC is axisymmetric rotation, the dipole should be on or parallel to the flight track. Carbone and Marks (1988) noted that the peak Doppler velocities of the dipole in Fig. 6 rotated counterclockwise from the flight track. They proposed that the azimuthal rotation of the dipole resulted in a radial inflow near the eyewall. The alignment of the Doppler velocity dipole relative to the flight track is affected by the direction of the radial flow (i.e. inflow or outflow) and the ratio of the magnitude of the radial flow to the tangential flow. Hence, the axi-symmetric tangential and radial winds can be estimated from the VTD.

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

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

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

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

The concept of the VTD technique is illustrated in Fig. 7, where the Doppler velocity (black vectors) of an axisymmetric rotation and inflow [gray vectors in Fig. 7(a) and 7(b)] plotted against the angle form a negative sine and cosine curve, respectively. The Doppler velocity pattern of a combined axisymmetric rotation and radial inflow is a negative sine curve with a negative (clockwise) phase shift [Fig. 7(c)]. It can be shown that the sign of the phase shift corresponds to the direction of the axisymmetric radial flow at that radius, while the magnitude of the phase shift is proportional to the ratio of the axisymmetric radial and the tangential velocities. By adding an environmental mean flow to the picture, the entire curve will be shifted vertically upward or downward, depending on aThe GBVTD formulation is a more general form of the VTD formulation (Lee et al., 1999; Jou et al., 2008). VTD is a special case of GBVTD when a vortex is located at an infinite distance from the radar.

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

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

the mean flow direction (i.e. the zero isodop will not bisect the dipole in an axisymmetric vortex). In fact, this simple illustration of the GBVTD concept summarizes the dipole signatures of an atmospheric vortex reviewed in Subsec. 2.2. Real atmospheric vortices also contain asymmetric structures; therefore, the resulting Doppler velocity profile is more complicated than that illustrated in Fig. 7(c). Different wave numbers can be distinguished via Fourier decomposition of the Doppler velocity profile along each radius in the vortex cylindrical coordinate. Therefore, the 3D vortex structure can be deduced by compositing GBVTD analyses at each radius and height, providing that the circulation center at each altitude is known accurately. The issues related to vortex center finding will be discussed in Sec. 4.

Owing to insufficient independent equations in the VTD formulation, Lee et al. (1994) and Lee et al. (1999) assumed that the asymmetric radial winds are much smaller than their tangential counterparts, and acknowledged that the cross-beam mean wind component cannot be resolved, thus letting radial winds and the cross-beam mean wind alias into tangential winds to close the system. Note that if the radar is located at the center of a vortex, then only the radial wind component can be measured in that situation, and GBVTD becomes VAD. When the vortex is located at an infinite distance (i.e. the aspect ratio is small for practical purposes), the radar beams can be assumed parallel and the GBVTD formulation reduced to a simpler VTD formulation. With the knowledge of axisymmetric tangential and radial winds at each radius and altitude, several additional kinematic (convergence and vertical velocity) and dynamic (angular momentum and pressure gradient) quantities can be computed (e.g. Lee et al., 2000; Lee and Wurman, 2005; Lee and Bell, 2007).

Sensitivity tests of the GBVTD technique using analytical vortices showed that accurate axisymmetric and good asymmetric vortex structures could be deduced from single Doppler observations. However, the distortion in phase and amplitude of asymmetric vortex structures worsens as the wave number increases (Lee et al., 1999). Note that the analytical vortices simulated by Lee et al. (1999) did not contain asymmetric radial winds; thus, the biases from ignoring asymmetric radial winds could not be evaluated. Nevertheless, Lee et al. (1999) demonstrated that the GBVTD technique could extract plausible kinematic and wave structures embedded in the Doppler velocities of atmospheric vortices.

The EVTD technique, proposed by Roux and Marks (1996), expanded the VTD analysis to successive radial passes (legs) in order to improve the stability of the VTD analysis and extract wave number 1 radial wind information. EVTD first solves for a constant mean wind vector by performing a VTD analysis with limited coefficients on each altitude. The mean wind vector is then removed, and the residual Doppler velocities are used to solve for the coefficients at all radii simultaneously by a least-squares minimization of a cost function. This cost function minimizes three constraints: (1) the differences between mean wind relative Doppler velocities and EVTD-derived radial velocities, (2) the random variation of coefficients between the rings, and (3) the mean wind residuals resulting from the wave number 1 asymmetry. The EVTD coefficients deduced in each pass are then filtered to obtain coherent mesoscale structures over time. A similar extension of the GBVTD technique, the GB-EVTD technique, was proposed by Roux et al. (2004) to process tropical cyclone data collected by a ground-based Doppler radar on successive scans.

4. Atmospheric Vortex Circulation Center Deduced from Single Doppler Radar Data

Atmospheric vortex centers have different definitions based on available measurements and their associated assumptions, including the geometric center, defined as the centroid of the eyewall radar reflectivity (Griffin et al., 1992); the dynamic center, defined as the minimum of the stream function, pressure, or geopotential height (e.g. Willoughby and Chelmow, 1982; Dodge et al., 1999); and the vorticity (or circulation) center, defined as the point that maximizes the eyewall vorticity or circulation (e.g. Willoughby, 1992; Marks et al., 1992; Lee and Marks, 2000). These centers are not necessarily collocated, and the uncertainties in estimating them, range from several kilometers to tens of kilometers in TCs (Willoughby and Chelmow, 1982; Lee and Marks, 2000; Harasti et al., 2004).

Uncertainties and inconsistencies of estimated TC centers on the order of 10 km may not affect operational track forecasts, but they critically affect computing TC wind fields in a cylindrical coordinate system (Willoughby, 1992; Lee and Marks, 2000). This includes partitioning vortex wind fields from a Cartesian coordinate system to a cylindrical coordinate system (Wil-loughby, 1992; Marks et al., 1992) and retrieving vortex wind fields from single Doppler radar data directly in cylindrical coordinates (Lee et al., 1994; Roux and Marks, 1996; Lee et al., 2000; Roux et al., 2004; Lee and Wurman, 2005). Lee and Marks (2000) demonstrated that a vortex circulation center needs to be accurate within 1 km on a 20 km radius of maximum wind (RMW), or 5% of the RMW, to keep the nominal error of the apparent wave number 1 less than 20% of the maximum axisymmetric tangential winds (MATWs: VT0-max) in the GBVTD analysis. Since the vorticity (circulation) center is directly related to the accuracy of the deduced vortex wind fields, it will be the focus of this section.

Harasti and List (2005) applied a principal component analysis (PCA) to the single Doppler velocity data arranged sequentially in a matrix according to the range and azimuth coordinates. For a Rankine vortex, the coordinates (location) of particular cusps in the curve of the first two eigenvector coefficients plotted against their indices are geometrically related to both the circulation center and the RMW. The uncertainty of the PCA method regarding asymmetric TCs is still a research topic.

The vorticity center can be computed from a given 2D horizontal vortex wind field and a simplex method (Neldar and Mead, 1965) to identify a point that maximizes the eyewall vor-ticity (Marks et al., 1992). However, estimating TC centers using this technique is not straightforward with a single Doppler radar, because the velocity patterns do not provide a complete 2D wind field for direct vorticity computation. Lee and Marks (2000) proposed a novel approach, and invented the GBVTD-simplex algorithm, by using the simplex method to ite-ratively identify a center that maximized the GBVTD-derived eyewall vorticity. The search process of the simplex algorithm toward the true center of a Rankine vortex centered at (0, 60), with an RMW of 20 km and a maximum tangential wind of 50ms-1, is illustrated in Fig. 8.

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

Figure 8. Illustration of the GBVTD-simplex convergence (thick dashed line) and the intermediate simplex from the initial guess at I ( — 10.0, 50.0). The triangle A—B—C is the initial simplex with a centroid at I. R, X, and C indicate the resulting points from reflection, expansion, and contraction operations with the number corresponding to the individual simplex. The TC center is located at (0.0, 60.0) with a hurricane symbol. Only the first six iterations are shown here. The GBVTD-derived axisymmetric tangential winds are listed in parentheses in m s~1. Differently shaped lines distinguish intermediate triangles. (From Lee and Marks, 2000.)

Figure 8. Illustration of the GBVTD-simplex convergence (thick dashed line) and the intermediate simplex from the initial guess at I ( — 10.0, 50.0). The triangle A—B—C is the initial simplex with a centroid at I. R, X, and C indicate the resulting points from reflection, expansion, and contraction operations with the number corresponding to the individual simplex. The TC center is located at (0.0, 60.0) with a hurricane symbol. Only the first six iterations are shown here. The GBVTD-derived axisymmetric tangential winds are listed in parentheses in m s~1. Differently shaped lines distinguish intermediate triangles. (From Lee and Marks, 2000.)

on the value of interest to determine the subsequent search toward the maximum or minimum value. In this example, the MATWs are computed starting from vertices A, B, and C as the vortex centers of GBVTD analyses. Vertex A has the lowest MATW value, yielding a reflection point of A (R1) on segment BC. Then the MATW using R1 as a center [MATW(R1)] is computed and compared with those MATWs in A, B, and C. Subsequent operations to determine a new vertex replacing A to form the next simplex are based on whether MATW(R1) is a maximum (expansion), intermediate (R1 replaces A), or minimum (contraction). The above processes are repeated until a convergence criterion is reached. The thick, dashed line (in Fig. 8) represents the converging path of intermediate TC centers toward the true center at (0, 60).

The simplex method described here is efficient in finding a local maximum or minimum within the parameters specified by the users. In practice, it is recommended to run the simplex algorithm from various initial guesses, preferably from all quadrants of the "true" center. Without independent verifications, the grouping or scattering of the simplex results (end points) is the only statistical indication of the quality (uncertainty) of the GBVTD-simplex-derived vortex center. The mean error in the GBVTD-simplex-derived vortex centers using analytical vortices is 0.35m (< 2% of a 20km RMW), which is significantly less than the 1 km radial grid spacing in the cylindrical coordinates. Random errors of 5ms-1 added to the Doppler velocity do not change the mean error and standard deviation much. When applied to real atmospheric vortices, the standard deviations increased to —2 km in Typhoon Alex (-9% of a 23 km RMW; Lee and Marks, 2000) and -30 m in the Mulhall tornado (-5% of a 600 m RMW; Lee and Wurman, 2005). The accuracy of the GBVTD-simplex algorithm strongly depends on the accuracy of the GBVTD-derived MATW. Since the MATW can be biased by the unknown cross-beam mean flow and the unresolved wave number 2 radial flow, the GBVTD-simplex algorithm has experienced difficulties in deducing centers in elliptical (wave number 2) vortices such as Typhoon Herb and Nari (Michael Bell, 2005; personal communication). Further quantifying the error characteristics of the GBVTD-simplex algorithm in real atmospheric vortices requires independent measurement of the "true" vortex center from either in situ measurements or a second set of Doppler radar observations from a different viewing angle.

5. Atmospheric Vortex Structures

### 5.1. Structure of tropical cyclones

The first physically consistent 3D axisymmetric kinematic structure of a TC (Hurricane Gloria, 1985) deduced from NOAA WP-3D single Doppler radar data and the VTD technique was given by Lee et al. (1994). Figure 9 illustrates axisymmetric structures of a TC previously only available from flight-level composite data (e.g. Jorgensen, 1984a,b), pseudo-dual Doppler analysis (e.g. Marks and Houze, 1987; Marks et al., 1992; Franklin et al., 1993), or numerical simulations (e.g. Rotunno and Emanuel, 1987). Key features in Fig. 9 include:

(1) An upright eyewall reflectivity (—R = 13 km) is accompanied by an MATW exceeding 60 ms-1 at an RMW of —18 km, while the axisymmetric tangential winds decrease rapidly both inside and outside the RMW [Fig. 9(a)]. The decrease of the MATW with height suggests a warm core structure from the thermal wind relationship.

(2) A shallow, low-level inflow (—1 km deep) inside R — 32 km peaks (> 3ms-1) as it approaches the eyewall (Fig. 9b). It converges near the eyewall [Fig. 9(c)] and turns into an updraft in the eyewall [> 4 ms-1; Fig. 9(d)].

(3) Immediately outside the eyewall, a region of downdraft (> 2ms-1, between R = 16 km and 40 km) is consistent with weak reflectivity and radial inflow beneath the anvil outflow [dashed lines between 8 km and 13 altitude in Fig. 9(b)]. This branch of inflow beneath the anvil was seldom resolved in previous pseudo-Doppler analysis but appeared in numerically simulated TCs [Fig. 5(c) in Rotunno and Emanuel, 1987].

The VTD-derived asymmetric structure of Gloria over two successive legs suggested that the hurricane evolved from a wave number 1 [Fig. 10(a)] to a wave number 2 [Fig. 10(b)] structure within an hour. A pseudo-dual Doppler (PDD) analysis using combined data from these two legs showed a wave number 3 structure [Fig. 10(c)], suggesting temporal aliasing in the pseudo-dual Doppler analysis. Again, the advantage of reducing the effective stationary time period from —1 hour (PDD) to —20 minutes (VTD) was clearly demonstrated.

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

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

Lee et al. (1994) derived a hodograph of the Earth-relative horizontal wind from these two legs of data (Fig. 11) and found that Gloria's mean motion vector is ^30° to the left of the deep-layer, density-weighted mean wind vector, consistent with those computed from the deep-layer mean wind in a much larger domain of several degrees in longitude and latitude (e.g. George and Gray, 1976; Chan and Gray, 1982; Holland, 1984; Dong and Neumann, 1986; Hanley et al., 2001).

The structure and evolution of Hurricane Hugo (1989) over a 7-hour period on 17 September 1989 were presented by Roux and Marks (1996) using EVTD-derived winds from six successive eye crossings about 70 minutes apart. The axisymmetric and asymmetric structures in Hugo were qualitatively consistent with those in Gloria, Alicia, and Norbert. More importantly, Roux and Marks (1996) demonstrated that plausible wave number 1 radial wind could be derived from the EVTD technique using multiple legs of airborne Doppler radar data. The EVTD-derived radial wind rotated from easterly at a 2 km altitude to southerly at 5 km and became southwesterly at 10 km, and compared favorably with the PDD-derived radial winds (Fig. 12).

The ability to deduce kinematic structures of landfalling TCs from coastal Doppler radar data has been demonstrated through Typhoon Alex (1987) (Lee et al., 2000), the concentric eyewall (double wind maxima) in Typhoon Billis (2000) (Jou et al., 2001), Hurricane Bret (1999) (Harasti et al., 2003), and Hurricane Charley (2004) (Lee and Bell, 2007) using the

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