Abstract

Radar echo images demonstrate that mature tropical cyclones frequently have a concentric eyewall structure, which consists of the inner eyewall, echo-free moat, and outer eyewall regions. Near the inner and outer eyewalls, well-defined wind maxima are generally observed. This indicates that two large vertical vorticity regions exist just inside radii of the two wind maxima near the inner and outer eyewalls. Therefore, the concentric eyewall structure can be considered to be a double vortex composed of the inner vortex and the outer vortex ring. In this study, the contour dynamics model is used on the f plane to analyze the characteristics of flows with either a symmetric double vortex or an asymmetric one, and examined the relationship between the movement of the inner vortex in an asymmetric double vortex and a trochoidal motion of a tropical cyclone with an asymmetric concentric eyewall structure.

Results show that, depending on the degree of an interaction of a double vortex, the evolution of the inner vortex is classified into three patterns: the first is that the center of the inner vortex is stationary, which is seen only for the symmetric double vortex; the second is that the track of the center of the inner vortex draws a circle; and the third is that it draws a spiral. A numerical experiment based on an observed flow around Typhoon Herb was also performed. The time evolution of the double vortex is very similar to that of radar echo intensity of Typhoon Herb. Also the rotation period and amplitude of the inner vortex in the numerical experiment were comparable with those of the trochoidal motion in the observation. These suggest that, in tropical cyclones with the concentric eyewall structure, the interaction of an asymmetric double vortex can become a cause of trochoidal motion.

1. Introduction

Radar reflectivities demonstrate that typical tropical cyclones with strong tangential wind have a symmetric eyewall structure and are accompanied by several spiral bands. Observational studies using aircraft have captured a characteristic structure that the spiral bands often generate a well-defined ring of heavy rainfall radially outside the primary eyewall and consequently form the secondary eyewall. This structure is generally called the concentric eyewall structure (Holliday 1977; Willoughby et al. 1982; Black and Willoughby 1992; Roux and Viltard 1995). As an example of tropical cyclones with the concentric eyewall structure, Fig. 1 shows a horizontal distribution of a composite radar reflectivity in Hurricane Edouard and a representative radial profile of tangential wind and vertical vorticity, which were measured by the National Oceanic and Atmospheric Administration (NOAA) WP-3D research aircraft at a flight level of 700 hPa on 29 August 1996. The concentric eyewall structure consists of the inner eyewall, echo-free moat, and outer eyewall regions (Fig. 1a). Figure 1b illustrates that wind maxima in the inner and outer eyewall regions existed near radii of 20 and 45 km, respectively. As a result, Hurricane Edouard had two large vorticity regions just inside radii of the two wind maxima. Tropical cyclones with the concentric eyewall structure usually form such a double vorticity distribution (e.g., Samsury and Zipser 1995; Dodge et al. 1999), and most of these tropical cyclones seem to have a symmetric structure in radar reflectivities.

Fig. 1.

(a) Horizontal distribution of a composite radar reflectivity in Hurricane Edouard on 29 Aug 1996. The domain size is 240 km × 240 km. The composite is produced from the NOAA WP-3D flight track data at 700 hPa from 2251 to 2321 UTC. (b) Profiles of tangential wind velocity (solid line) and vertical vorticity (dashed line) at 700 hPa during one leg from 2309 to 2333 UTC.

Fig. 1.

(a) Horizontal distribution of a composite radar reflectivity in Hurricane Edouard on 29 Aug 1996. The domain size is 240 km × 240 km. The composite is produced from the NOAA WP-3D flight track data at 700 hPa from 2251 to 2321 UTC. (b) Profiles of tangential wind velocity (solid line) and vertical vorticity (dashed line) at 700 hPa during one leg from 2309 to 2333 UTC.

Tropical cyclones with the concentric eyewall structure are often accompanied by trochoidal motion (Jordan 1966; Muramatsu 1986). Jordan (1966) examined Hurricane Carla and its trochoidal motion using a land-based radar. He pointed out that the position of the inner vortex did not always coincide with the geometric center of the outer vortex ring; that is, Hurricane Carla had an asymmetric, concentric eyewall structure. He also suggested that a rotary motion of the inner vortex around the geometric center of the outer vortex ring caused a trochoidal motion of Hurricane Carla, and that the amplitude of the trochoidal motion was within a range of the core region of Hurricane Carla. A more detailed radar echo pattern of a trochoidal motion was observed in Typhoon Wynne during its passage around the Miyakojima Island (Muramatsu 1986). The radar images demonstrated that Typhoon Wynne had an asymmetric, concentric eyewall structure and its center moved trochoidally. Muramatsu (1986) pointed out that the trochoidal motion occurred while Typhoon Wynne had the asymmetric, concentric eyewall structure, and that the movement of the inner eye depended on the rotation of the outer eyewall.

A trochoidal motion was found also in Typhoon Herb (Itano et al. 2002). Using a linearized model, Oda et al. (2005) attempted to explain the trochoidal motion as a wavenumber-1 perturbation arising from an instability of a symmetric vortex; however, the period of the wavenumber-1 perturbation was considerably smaller than that of the observed trochoidal motion. Actually, Typhoon Herb also had an asymmetric, concentric eyewall structure while it moved trochoidally.

The above studies seem to suggest that, in tropical cyclones with the concentric eyewall, only the asymmetric structure causes trochoidal motion. The purpose of this study is to simulate the movement of the inner vortex in flows with either a symmetric double vortex or an asymmetric one, and to examine the relationship between its movement and the trochoidal motion. For this purpose, we perform more than 140 numerical experiments with the two-dimensional contour dynamics model (e.g., Dritschel 1985, 1988, 1989).

The outline of this paper is as follows. The numerical model used in this study is summarized in section 2. Results of numerical experiments on the interaction of a double vortex in various flow regimes are presented in section 3. In section 4, a numerical experiment based on the flow around Typhoon Herb is done, and the time evolution, rotation amplitude, and rotation period of the inner vortex in the numerical experiment are compared with those in the observation. Sections 5 and 6 provide discussions and conclusions of this study.

2. The contour dynamics model

Various physical processes, such as the moist convection and the surface friction, contribute to the formation of tropical cyclones. Since these processes may obscure the fundamental mechanism of trochoidal motion, however, we neglect them and consider the conservative barotropic nondivergent dynamics. In this study, we use the contour dynamics model (e.g., Dritschel 1985, 1988, 1989), which assumes a piecewise vorticity field and calculates the time change of the field with a Lagrangian method. Because the vorticity peak in the core of tropical cyclones is much higher than that in the surrounding flow, a piecewise vortex patch is considered to be the first approximation of continuous vorticity profiles of tropical cyclones (Ritchie and Holland 1993; Prieto et al. 2003).

Contours are shaped to boundaries of vortex patches and each contour is divided into N nodes. In a contour, the velocity field U is expressed as

 
formula

where

 
formula

M is the number of contours, k the vortex jump across the kth contour, and Ck the kth contour, and Xj = (xj, yj) the position vector. The contour integration is done counterclockwise along the contour Ck. The time evolution of the position vector Xj is defined as

 
formula

In this study, we idealize the concentric eyewall structure and initially give two vortex patches with uniform intensities of vertical vorticity. The calculation is done on the f plane; M and N are set to 3 and 90, respectively. The computational domain is unbounded and the flow is assumed to be irrotational at infinity. Time integration is done using a standard second-order Runge–Kutta scheme with a nondimensional time step of 5 × 10−3.

3. Interaction of a double vortex in various flow regimes

In this section, we perform a total of 140 numerical experiments to examine the interaction of a double vortex in tropical cyclones with an idealized, concentric eyewall structure. The system is nondimensionalized. As typical values for tropical cyclone, the horizontal length scale L and the velocity scale V are selected to be 100 km and 20 m s−1, respectively. Then the time scale L/V leads to 5 × 103 s. Figure 2 shows an example of a radial profile of a double vortex for a symmetric, concentric eyewall structure. To represent the existence of the inner eyewall (closed vortex) and the outer eyewall (vortex ring), we set the inner vorticity ζi = 2.0 at r = 0.0–0.2 and the outer vorticity ζo = 1.0 at r = 1.0–1.2, where r is the nondimensional distance from the center of the outer vortex ring. Numerical experiments were performed by changing the r coordinate Ci of the center of the inner vortex from 0.0 to 0.8 with an interval of 0.1, the radius ri of the inner vortex from 0.1 to 0.5 with an interval of 0.1, and the ratio of the outer vorticity to the inner vorticity, ζo/ζi, from 0.1 to 2.0. Here, the intensity of ζi and the width of ζo are fixed at 2.0 and 0.2, respectively.

Fig. 2.

Radial profile of a normalized double vorticity in a concentric eyewall structure, which is idealized as the initial state of the 2D contour dynamics model. The inner vorticity ζi is set to 2.0 at r = 0.0 − 0.2, the moat region vorticity ζm to 0.0 at r = 0.2 − 1.0, and the outer vorticity ζo to 1.0 at r = 1.0 − 1.2.

Fig. 2.

Radial profile of a normalized double vorticity in a concentric eyewall structure, which is idealized as the initial state of the 2D contour dynamics model. The inner vorticity ζi is set to 2.0 at r = 0.0 − 0.2, the moat region vorticity ζm to 0.0 at r = 0.2 − 1.0, and the outer vorticity ζo to 1.0 at r = 1.0 − 1.2.

Figure 3 shows regime diagrams of evolution patterns for symmetric (Ci = 0.0) and asymmetric (Ci > 0.0) double vortices as functions of Ci (horizontal axis) and ri (vertical axis). Figures 3a–d show results from four cases of ζo/ζi = 0.1, 0.5, 1.0, and 2.0, respectively. The movement of the center of the inner vortex varies according to the degree of an interaction with the outer vortex ring. This movement can be classified into three patterns: stationary, circular, and spiral, where the spiral pattern refers to the case in which after one rotation around the center of the outer vortex ring, the center position of the inner vortex deviates from its initial center position by more than 5% of Ci. The stationary pattern is seen only for symmetric double vortices and the spiral pattern decreases as the vorticity ratio ζo/ζi increases.

Fig. 3.

Regime diagrams for symmetric and asymmetric double vortices as functions of the normalized center position Ci and radius ri of the inner vortex: (a) ζo/ζi = 0.1, (b) ζo/ζi = 0.5, (c) ζo/ζi = 1.0, and (d) ζo/ζi = 2.0, where ζo/ζi is the ratio of the outer vorticity to the inner vorticity. The stationary pattern refers to the regime that a double vortex maintains a symmetric structure, the circular pattern to the regime that the inner vortex circularly rotates around the center of the outer vortex ring, and the spiral pattern to the regime that the movement of the inner vortex is spiral.

Fig. 3.

Regime diagrams for symmetric and asymmetric double vortices as functions of the normalized center position Ci and radius ri of the inner vortex: (a) ζo/ζi = 0.1, (b) ζo/ζi = 0.5, (c) ζo/ζi = 1.0, and (d) ζo/ζi = 2.0, where ζo/ζi is the ratio of the outer vorticity to the inner vorticity. The stationary pattern refers to the regime that a double vortex maintains a symmetric structure, the circular pattern to the regime that the inner vortex circularly rotates around the center of the outer vortex ring, and the spiral pattern to the regime that the movement of the inner vortex is spiral.

Figure 4 shows contour plots of the vorticity field in a numerical experiment of the evolution of a symmetric double vortex with Ci = 0.0, ri = 0.2, and ζo/ζi = 0.5 (see Fig. 2). The center of the inner vortex does not move from the origin with time and the double vortex maintains its symmetric structure. It is therefore considered that trochoidal motion cannot occur in tropical cyclones with a two-dimensional, symmetric, concentric eyewall structure. However, note that we take no account of the growth of instabilities that may cause an asymmetric structure and the resulting trochoidal motion.

Fig. 4.

Contour plots of the vorticity field for a numerical simulation of a double vortex in the stationary pattern: (a) t = 0 and (b) t = 50. The center position Ci and radius ri of the inner vortex are equal to 0.0 and 0.2, respectively, and the vorticity ratio ζo/ζi is equal to 0.5. (c) Time change of the center of the inner vortex in the y direction.

Fig. 4.

Contour plots of the vorticity field for a numerical simulation of a double vortex in the stationary pattern: (a) t = 0 and (b) t = 50. The center position Ci and radius ri of the inner vortex are equal to 0.0 and 0.2, respectively, and the vorticity ratio ζo/ζi is equal to 0.5. (c) Time change of the center of the inner vortex in the y direction.

The circular pattern is observed over a wide range of our experimental parameters representing an asymmetric double vortex. Contour plots of the vorticity field in a numerical experiment of the evolution of the circular pattern with Ci = 0.2, ri = 0.2, ζo/ζi = 0.5 are shown in Figs. 5a–d. The inner vortex rotates cyclonically around the center of the outer vortex ring. Because of an interaction of the double vortex, a wavelike oscillation also starts to occur inside the outer vortex ring (Fig. 5d). The center position of the inner vortex in the y direction is shown as a function of time in Fig. 5e. This illustrates that both the rotation period and amplitude of the center of the inner vortex change very little with time. This steady rotation is clearly seen in the trajectory of the center of the inner vortex (Fig. 5f).

Fig. 5.

Contour plots of the vorticity field for a numerical experiment of an asymmetric double vortex in the circular pattern: (a) t = 0, (b) t = 10, (c) t = 30, and (d) t = 50. Values of Ci, ri, and ζo/ζi are 0.2, 0.2, and 0.5, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

Fig. 5.

Contour plots of the vorticity field for a numerical experiment of an asymmetric double vortex in the circular pattern: (a) t = 0, (b) t = 10, (c) t = 30, and (d) t = 50. Values of Ci, ri, and ζo/ζi are 0.2, 0.2, and 0.5, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

On the other hand, the spiral pattern appears when ζo/ζi is small and the spacing between the outer edge of the inner vortex and the inner edge of the vortex ring is small (see Fig. 3). Contour plots of the vorticity field in a numerical experiment of the evolution of the spiral pattern with Ci = 0.6, ri = 0.3, and ζo/ζi = 0.5 are shown in Figs. 6a–d. The inner vortex rotates cyclonically and the inside of the outer vortex ring starts to oscillate after t = 2. Unlike the circular pattern, however, a filament structure growing from the inside of the outer vortex ring winds around the inner vortex (Fig. 6d). Similar filament structures were discussed by Ritchie and Holland (1993) and Prieto et al. (2003). They examined the interaction between two closed vortices through numerical experiments, and demonstrated that the filament structure from one vortex wound around the other vortex according to the vorticity ratio of the two vortices and their spacing. The y coordinate of the center of the inner vortex and its trajectory are shown in Figs. 6e and 6f, respectively. Obviously, the center of the inner vortex does not draw a circle. In brief, the spiral pattern implies the occurrence of the filament structure. Since the spiral pattern tends to appear for small ζo/ζi, the filament structure is considered to suggest that the outer vortex ring is not very strong compared with the inner vortex.

Fig. 6.

Same as Fig. 5 but for the spiral pattern: (a) t = 0, (b) t = 2, (c) t = 6, and (d) t = 10. Values of Ci, ri, and ζo/ζi are 0.6, 0.3, and 0.5, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

Fig. 6.

Same as Fig. 5 but for the spiral pattern: (a) t = 0, (b) t = 2, (c) t = 6, and (d) t = 10. Values of Ci, ri, and ζo/ζi are 0.6, 0.3, and 0.5, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

Figure 7 shows the dependence of the rotation period and amplitude of the center of the inner vortex in the circular pattern on both the vorticity ratio ζo/ζi and the center position Ci of the inner vortex, where Ci changes from 0.2 to 0.6 and the radius ri of the inner vortex is fixed at 0.2. The rotation period of the inner-vortex center decreases with increasing ζo/ζi (Fig. 7a), implying that its rotation becomes fast as ζo increases because of the fixed ζi. However, its period changes very little with Ci. It is found that, for a given ζi, the rotation period of the inner-vortex center depends only on ζo/ζi. On the other hand, although the amplitude of the inner-vortex center changes very little with ζo/ζi, it increases with increasing Ci. Therefore, the amplitude of the inner-vortex center depends only on Ci.

Fig. 7.

(a) Rotation period of the inner vortex and (b) its amplitude as a function of the vorticity ratio ζo/ζi. The radius ri of the inner vortex is fixed at 0.2 and its center position Ci is set to 0.2, 0.4, and 0.6.

Fig. 7.

(a) Rotation period of the inner vortex and (b) its amplitude as a function of the vorticity ratio ζo/ζi. The radius ri of the inner vortex is fixed at 0.2 and its center position Ci is set to 0.2, 0.4, and 0.6.

Finally, we examine the effect of the vorticity ζm in the moat region, which is between the outer edge of the inner vortex and the inner edge of the outer vortex ring, by setting ζm to more than zero. Figure 8 shows the time evolution of the amplitude of the inner-vortex center as a function of time, where ζm is set to 0.1, 0.5, and 0.9. It is found that the rotation period of the inner-vortex center decreases with increasing ζm. This may be because the increase of ζm corresponds to the relative decrease of ζi (see Fig. 7a). Meanwhile, the amplitude of the inner-vortex center varies very little with ζm. Thus ζm has only an effect of accelerating the rotation of the inner-vortex center.

Fig. 8.

Time change of the y coordinate of the center of the inner vortex for ζm= 0.1, 0.5, and 0.9. The initial state of a double vortex is the same as in Fig. 5.

Fig. 8.

Time change of the y coordinate of the center of the inner vortex for ζm= 0.1, 0.5, and 0.9. The initial state of a double vortex is the same as in Fig. 5.

4. Comparison with a trochoidal motion of Typhoon Herb

In the previous section, we presented the time evolution of a double vortex in various flow regimes and the characteristics of the rotation period and amplitude of the center of the inner vortex. These results demonstrated that the inner vortex rotated cyclonically around the center of the outer vortex ring. This behavior is similar to trochoidal motion of tropical cyclones with an asymmetric, concentric eyewall structure. Typhoon Herb also had an asymmetric, concentric eyewall structure. Oda et al. (2005) inferred that a trochoidal motion of Typhoon Herb was not a wavenumber-1 perturbation arising from an instability of the symmetric flow. Its trochoidal motion may be therefore explained as the rotation of the inner vortex due to the interaction of an asymmetric double vortex. In this section, we first perform a numerical experiment based on an observed flow around Typhoon Herb. After that, we compare the time evolution of the asymmetric double vortex and the rotation period and amplitude of the inner vortex with the observations.

Figure 9 shows the track of the center of Typhoon Herb on 30–31 July 1996. Its track is the best track, which is determined by the Japan Meteorological Agency on the basis of satellite and radar images and meteorological data. In Fig. 9, it seems that Typhoon Herb began to move trochoidally at about 1500 Japanese standard time (JST) on 30 July 1996. If one cycle of the trochoidal motion is considered to be from 0600 to 1800 JST on 31 July 1996 when the typhoon was closest to the Sakishima Islands (the dashed line in Fig. 9), its period is 12 h. Its amplitude, on the other hand, is estimated to be about 20 km, as a half of the fluctuation from a straight line by which the two points at 0600 and 1800 JST are joined. Figure 10 shows horizontal distributions of radar echo images of Typhoon Herb observed by the Ishigaki radar during its trochoidal motion. These radar images illustrate that Typhoon Herb had an asymmetric, concentric eyewall structure, and that the inner eyewall rotated cyclonically around the center of the outer eyewall. A half cycle of its rotation took about 6 h, corresponding with that of the trochoidal motion (Fig. 9). Obviously, this suggests that the rotation of the inner eyewall was associated with the trochoidal motion. After 1500 JST, a filament structure from the outer eyewall was clearly seen.

Fig. 9.

Track of Typhoon Herb during its passage around the Sakishima Islands on 30–31 Jul 1996. (bottom) Lines and crosses indicate the typhoon track and the center position, respectively. In particular, dashed line shows the trochoidal path used to estimate the amplitude and period of oscillation. The best track data, which are determined from satellite and radar images and meteorological data, are provided by the Japan Meteorological Agency.

Fig. 9.

Track of Typhoon Herb during its passage around the Sakishima Islands on 30–31 Jul 1996. (bottom) Lines and crosses indicate the typhoon track and the center position, respectively. In particular, dashed line shows the trochoidal path used to estimate the amplitude and period of oscillation. The best track data, which are determined from satellite and radar images and meteorological data, are provided by the Japan Meteorological Agency.

Fig. 10.

Horizontal distributions of radar echo images of Typhoon Herb observed by the Ishigaki radar at (a) 1100, (b) 1300, (c) 1500, and (d) 1700 JST on 31 Jul 1996. The domain size is 500 km × 500 km.

Fig. 10.

Horizontal distributions of radar echo images of Typhoon Herb observed by the Ishigaki radar at (a) 1100, (b) 1300, (c) 1500, and (d) 1700 JST on 31 Jul 1996. The domain size is 500 km × 500 km.

In this study, we used the surface wind data at the Yonaguni and Iriomote Observatories of the Japan Meteorological Agency to estimate the distribution of the vertical vorticity around Typhoon Herb. The time series of the surface wind data every 10 min were transformed into a spatial distribution according to the distance from the center of the typhoon moving continuously. Figures 11a,b show the resulting radial profiles of the vertical vorticity arranged according to the distance from the centers of the inner and outer eyewalls, respectively. The profiles are normalized by the maximum wind of 35 m s−1 and its radius of 35 km. Figure 11a illustrates that the vorticity inside the inner eyewall has a maximum of ζ ≈ 3.0 near r ≈ 0.7 and tends to decrease as the center of the inner eyewall is approached. Thus the large vorticity region inside the inner eyewall is like a ring. However, we consider this region to be in the shape of a disk, for simplicity, to eliminate possibly dominant instabilities that may occur across the inner eyewall (e.g., Kossin et al. 2000). The vorticity ζi and radius ri of the disk-like inner vortex are assumed to be 3.0 and 0.7, respectively, which nearly correspond with the maximum vorticity and its position, respectively. As shown in Fig. 10, the inner vortex moves inside the outer eyewall. This behavior of the inner vortex can be seen within r ≈ 1.5 from the center of the outer eyewall (Fig. 11b), which nearly coincides with the diameter, 2 × ri, of the inner vortex. The center position Ci of the inner vortex can be therefore assumed to be nearly equal to ri = 0.7.

Fig. 11.

Radial profiles of the vertical vorticity of Typhoon Herb observed at the Iriomote and Yonaguni Observatories, which are normalized by the maximum wind of 35 m s−1 and its radius of 35 km: (a) the vertical vorticity arranged according to the distance from the center of the inner eyewall and (b) that from the center of the outer eyewall. Radial profile of (c) the vertical vorticity idealized as an asymmetric double vortex, where, based on the observations, the center position Ci and radius ri of the inner vortex are taken to be 0.7 and 0.7, and the vertical vorticities ζi, ζm, and ζo are set to 3.0 at r = 0.1–1.4, 0.0 at r = 1.4–2.1, and 0.4 at r = 2.1–2.7, respectively.

Fig. 11.

Radial profiles of the vertical vorticity of Typhoon Herb observed at the Iriomote and Yonaguni Observatories, which are normalized by the maximum wind of 35 m s−1 and its radius of 35 km: (a) the vertical vorticity arranged according to the distance from the center of the inner eyewall and (b) that from the center of the outer eyewall. Radial profile of (c) the vertical vorticity idealized as an asymmetric double vortex, where, based on the observations, the center position Ci and radius ri of the inner vortex are taken to be 0.7 and 0.7, and the vertical vorticities ζi, ζm, and ζo are set to 3.0 at r = 0.1–1.4, 0.0 at r = 1.4–2.1, and 0.4 at r = 2.1–2.7, respectively.

Figure 11b shows that several regions of the large vorticity exist radially outside r = 2.0, around which the vorticity nearly vanishes. These regions are considered to be associated with the outer eyewall and rainbands. Since we aim to examine the interaction between the inner and outer vortices, we will take account of only the vortex closest to the inner vortex. Then the relevant outer vortex exists with ζo ≈ 0.4 at r = 2.1 − 2.7.

Assembling these results, the idealized distribution of the asymmetric double vortex is determined as Fig. 11c. This distribution was used as the initial condition for the numerical experiment. Since the vorticity ratio ζo/ζi is about 0.133 and the inner and outer vortices are very close, it is supposed that the evolution pattern of the vorticity field in Typhoon Herb was the spiral pattern (see Fig. 3).

The simulated evolution of the vorticity field is shown in Fig. 12. From the beginning of the experiment, the inner vortex starts to rotate cyclonically around the center of the outer vortex ring. As time elapses, a vorticity filament is extracted from the inside of the outer vortex ring toward the inner vortex. This filament structure is similar to the filament-like structure of the heavy rainfall region in the radar images (see Fig. 10). It is considered that the heavy rainfall region visualizes the filament structure of the vorticity field.

Fig. 12.

Contour plots of the vorticity field for the observed flow around Typhoon Herb: (a) t = 0, (b) t = 3, (c) t = 9, and (d) t = 13. The initial state of the double vortex is shown in Fig. 11c. Values of Ci, ri, and ζo/ζi are 0.7, 0.7, and 0.133, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

Fig. 12.

Contour plots of the vorticity field for the observed flow around Typhoon Herb: (a) t = 0, (b) t = 3, (c) t = 9, and (d) t = 13. The initial state of the double vortex is shown in Fig. 11c. Values of Ci, ri, and ζo/ζi are 0.7, 0.7, and 0.133, respectively. (e) Time change of the center of the inner vortex in the y direction. (f) Trajectory of the center of the inner vortex.

Since the movement of the inner vortex finishes about 1/2 cycles at t = 10.25 (Figs. 12e,f), the rotation period of the inner vortex is estimated to be about 20.5 in the nondimensional time and 5.7 h in the dimensional time. In Oda et al. (2005), the velocity of a depth average wind over the troposphere is evaluated to be 1.56 times larger than that of the surface wind. Considering this ratio, the rotation period results in 3.7 h. These rotation periods of 5.7 and 3.7 h are less than half the observed period of about 12 h of the trochoidal motion (see Fig. 9). The simulated rotation period is largely dependent on ζo/ζi. Figure 7a means that the smaller ζo becomes, the longer the rotation period becomes. In fact, it increases to about 1.2 times for ζo = 0.3, which is a possible value (see Fig. 11b). Therefore it may be thought that the rotation period of the inner vortex is roughly comparable with that of the trochoidal motion.

We also compare the amplitude of the inner vortex with the observations. The amplitude of the inner vortex obtained from the numerical experiment is somewhat smaller than Ci = 0.7 (25 km in the dimensional scale), where its initial amplitude was set to Ci = 0.7 (see Fig. 11c). This amplitude is in good agreement with the observed amplitude of about 20 km of the trochoidal motion (see Fig. 9). These results may suggest that the trochoidal motion of Typhoon Herb is caused by the interaction of an asymmetric double vortex.

5. Discussion

Radar echo images illustrated that Typhoon Wynne had an asymmetric, concentric eyewall structure and a filament structure elongated from the outer eyewall toward the inner eyewall (Muramatsu 1986). These characteristics are very similar to those of Typhoon Herb. Muramatsu (1986) claimed that the periodic oscillation of the track of Typhoon Wynne was due to the rotation of the inner eyewall around the geometric center of the outer eyewall. Our numerical simulation of Typhoon Herb also reached a similar conclusion by comparing with the radar images. Therefore, trochoidal motion of tropical cyclones with an asymmetric, concentric eyewall structure, such as Typhoons Wynne and Herb, is likely to be caused by the interaction of an asymmetric double vortex. However, we need to confirm the correspondence between the vorticity field and the heavy rainfall region.

Mechanisms different from the present mechanism have been discussed in various scales of trochoidal motion (e.g., Flatau and Stevens 1993; Holland and Lander 1993; Jones 1995; Wu and Emanuel 1993; Nolan et al. 2001; Itano and Ishikawa 2002). For example, Holland and Lander (1993) mentioned that the convective system near the primary vortex of Typhoon Sarah affected the track of its center. The amplitude and period of this trochoidal motion were about 100 km and a few days, respectively. On the other hand, Nolan et al. (2001) pointed out that an algebraic wavenumber-1 instability could spin the relatively low-vorticity core near the center of the primary eyewall. The amplitude and period of this trochoidal motion were very small in comparison with the other trochoidal motions. A possible future study would be to examine how many trochoidal motions can be caused by the present mechanism.

6. Conclusions

To examine the fundamental mechanism of trochoidal motion that has been often observed in tropical cyclones with an asymmetric, concentric eyewall structure, we performed numerical experiments using the contour dynamics model (e.g., Dritschel 1985, 1988, 1989). In this study, a concentric eyewall structure was idealized as a double-vortex structure composed of the inner closed vortex and the outer vortex ring. Results show that the evolution of a double vortex is classified into three patterns: the first is the stationary pattern in which a double vortex maintains its symmetric structure, the second is the circular pattern in which the inner vortex rotates cyclonically around the center of the outer vortex ring, and the third is the spiral pattern in which the inner vortex rotates cyclonically but its track is spiral. In addition, a significant characteristic of the spiral pattern is that a filament structure is formed from the inside of the outer vortex ring toward the inner vortex.

We also performed a numerical experiment based on an observed flow around Typhoon Herb, which had an asymmetric, concentric eyewall structure. The inner vortex rotates cyclonically around the center of the outer vortex ring, and a filament structure from the inside of the outer vortex ring appears gradually. This time evolution of the double vortex is very similar to that of radar echo intensity of Typhoon Herb. Also the rotation period and amplitude of the inner vortex in the numerical experiment were comparable with those of the trochoidal motion in the observation. These suggest that, in tropical cyclones with the concentric eyewall structure, the interaction of an asymmetric double vortex can become a cause of trochoidal motion.

Acknowledgments

We would like to express our appreciation to Professors H. Ueda and S. Yoden of Kyoto University, Dr. T. Iwayama of Kobe University, and Professor K. Tomine of National Defense Academy, for their valuable discussions and suggestions. We are also grateful to the Okinawa Observatory of the Japan Meteorological Agency for providing the observational data and to the NOAA Hurricane Research Division for providing the flight-level data.

REFERENCES

REFERENCES
Black
,
M. L.
, and
H. W.
Willoughby
,
1992
:
The concentric eyewall cycle of Hurricane Gilbert.
Mon. Wea. Rev.
,
120
,
947
957
.
Dodge
,
P.
,
R. W.
Burpee
, and
F. D.
Marks
Jr.
,
1999
:
The kinematic structure of a hurricane with sea level pressure less than 900 mb.
Mon. Wea. Rev.
,
127
,
987
1004
.
Dritschel
,
D. G.
,
1985
:
The stability and energetics of corotating uniform vortices.
J. Fluid Mech.
,
157
,
95
134
.
Dritschel
,
D. G.
,
1988
:
Contour surgery: A topological reconnection scheme for extended integrations using contour dynamics.
J. Comput. Phys.
,
77
,
240
266
.
Dritschel
,
D. G.
,
1989
:
Contour dynamics and contour surgery: Numerical algorithms for extended, high-resolution modeling of vortex dynamics in two-dimensional, inviscid, incompressible flows.
Comput. Phys. Rep.
,
10
,
77
146
.
Flatau
,
M.
, and
D. E.
Stevens
,
1993
:
The role of outflow-layer instabilities in tropical cyclone motion.
J. Atmos. Sci.
,
50
,
1721
1733
.
Holland
,
G. J.
, and
M.
Lander
,
1993
:
The meandering nature of tropical cyclone tracks.
J. Atmos. Sci.
,
50
,
1254
1266
.
Holliday
,
C. R.
,
1977
:
Double intensification of Typhoon Gloria, 1974.
Mon. Wea. Rev.
,
105
,
523
528
.
Itano
,
T.
, and
H.
Ishikawa
,
2002
:
Effect of negative vorticity on the formation of multiple structure of natural vortices.
J. Atmos. Sci.
,
59
,
3254
3262
.
Itano
,
T.
,
G.
Naito
, and
M.
Oda
,
2002
:
Analysis of elliptical eye of Typhoon Herb (T9609) (in Japanese with English abstract).
Sci. Eng. Rep. Natl. Def. Acad.
,
39
,
9
17
.
Jones
,
S. C.
,
1995
:
The evolution of vortices in vertical shear. Part I: Initially barotropic vortices.
Quart. J. Roy. Meteor. Soc.
,
121
,
821
851
.
Jordan
,
C. L.
,
1966
:
Surface pressure variations at coastal stations during the period of irregular motion of Hurricane Carla of 1961.
Mon. Wea. Rev.
,
94
,
454
458
.
Kossin
,
J. P.
,
W. H.
Schubert
, and
M. T.
Montgomery
,
2000
:
Unstable interactions between a hurricane’s primary eyewall and a secondary ring of enhanced vorticity.
J. Atmos. Sci.
,
57
,
3893
3917
.
Muramatsu
,
T.
,
1986
:
Trochoidal motion of the eye of Typhoon 8019.
J. Meteor. Soc. Japan
,
64
,
259
272
.
Nolan
,
D. S.
,
M. T.
Montgomery
, and
L. D.
Grasso
,
2001
:
The wavenumber-one instability and trochoidal motion of hurricane-like vortices.
J. Atmos. Sci.
,
58
,
3243
3270
.
Oda
,
M.
,
T.
Itano
,
G.
Naito
,
M.
Nakanishi
, and
K.
Tomine
,
2005
:
Destabilization of the symmetric vortex and the formation of the elliptical eye of Typhoon Herb.
J. Atmos. Sci.
,
62
,
2965
2976
.
Prieto
,
R.
,
B. D.
McNoldy
,
S. R.
Fulton
, and
W. H.
Schubert
,
2003
:
A classification of binary tropical-cyclone-like vortex interactions.
Mon. Wea. Rev.
,
131
,
2656
2666
.
Ritchie
,
E. A.
, and
G. J.
Holland
,
1993
:
On the interaction of tropical-cyclone-scale vortices. II: Discrete vortex patches.
Quart. J. Roy. Meteor. Soc.
,
119
,
1363
1379
.
Roux
,
F.
, and
N.
Viltard
,
1995
:
Structure and evolution of Hurricane Claudette on 7 September 1991 from air Doppler radar observation. Part I: Kinematics.
Mon. Wea. Rev.
,
123
,
2611
2639
.
Samsury
,
C. E.
, and
E. J.
Zipser
,
1995
:
Secondary wind maxima in hurricanes: Airflow and relationship to rainbands.
Mon. Wea. Rev.
,
123
,
3502
3517
.
Willoughby
,
H. E.
,
J. A.
Clos
, and
M. G.
Shoreibah
,
1982
:
Concentric eye walls, secondary wind maxima, and the evolution of the hurricane vortex.
J. Atmos. Sci.
,
39
,
395
411
.
Wu
,
C-C.
, and
K. A.
Emanuel
,
1993
:
Interaction of a baroclinic vortex with background shear: Application to hurricane movement.
J. Atmos. Sci.
,
50
,
62
76
.

Footnotes

Corresponding author address: Dr. Masahito Oda, Weather Center, Air Weather Group, Japan Air Self Defense Force, 1-5-5 Sengen-cho, Fuchu, Tokyo 183-8521, Japan. Email: hiro-ayu@f2.dion.ne.jp