1. Introduction

Axially asymmetric processes in the inner cores of tropical cyclones (TCs) affect the overall structures and time evolutions of TCs. Extensive theoretical and numerical studies have been conducted on the barotropic instabilities and resultant structural changes of the inner cores, which include intensification and the formation of polygonal eyewalls (e.g., Schubert et al. 1999; Kossin et al. 2000; Nolan and Montgomery 2000; Kossin and Eastin 2001). Observationally, such processes can be studied with radars and satellites (e.g., Reasor et al. 2000). However, observations of the dynamical processes further inside the eyes are limited because of the lack of precipitating particles, so previous studies use visual comparisons between numerical simulations and satellite imagery of lower clouds in the eyes (e.g., Kossin et al. 2002).

Inner cores of TCs also exhibit stable asymmetry. Environmental shear is known to incline the inner cores, which can rotate as vortex Rossby waves (Jones 1995; Reasor and Montgomery 2001). Storm translation induces stationary asymmetry in the boundary layer (Shapiro 1983; Kepert 2010). This is due to surface friction; without it, storm translation has no effect on the structure of vortices because of Galilean invariance. Observationally, stationary features can be studied statistically with aircraft observations (Zhang et al. 2013; Uhlhorn et al. 2014).

Observations by geostationary meteorological satellites are very important to monitor TCs, as they are used to estimate their intensity and tracks (Dvorak 1984). The satellites are also used to derive atmospheric motion vectors (AMVs), which are used in objective analyses to facilitate numerical forecasts of TCs. AMVs have been used to study TCs (e.g., Molinari and Vollaro 1989; Velden and Sears 2014; Oyama et al. 2016; Oyama 2017; Velden et al. 2017; Sawada et al. 2019; Fukuda et al. 2020). However, to the authors’ knowledge, there are no published studies on TC inner cores based mainly on AMVs.

The latest “third-generation” geostationary satellites Himawari-8 and GOES-R have imagers with higher resolutions than the satellites of the previous generation: 2 km at infrared (IR) and 0.5–1 km at visible wavelengths (Bessho et al. 2016; Schmit et al. 2017). Also, they support high-frequency imaging for specified subregions within the full disk. Himawari-8 is regularly operated to conduct imaging over a 1000 km × 1000 km region around tropical storms and typhoons at the time interval of 2.5 min, which is called the target observation of typhoons. From these high spatial and temporal resolutions, their data are useful to study TCs beyond conventional applications. By analyzing the IR brightness temperature from the target observation by Himawari-8, Horinouchi et al. (2020a) discovered gravity waves emanating concentrically from convective bursts in a form consistent with internal bore, and they suggested that to quantify the wave features would help estimate mass flux associated with convective bursts. Tsukada and Horinouchi (2020, hereinafter, TH20) used visible imagery to estimate the rotation velocity from the motion of clouds in the boundary layer of the mature eye of Typhoon Lan (2017); their method uses two-dimensional spectra along azimuth and time to find a representative rotational velocity at each radius. Tsujino et al. (2021) applied this method to IR imagery and successfully quantified the slowdown of the rotation in the eye of Typhoon Trami (2018).

Like their predecessors, the third-generation geostationary satellites are operationally used to derive AMVs. Stettner et al. (2019) described high-resolution AMVs derived from 1-min imaging of hurricanes by GOES-R, and the data are used in hurricane forecasts (e.g., Li et al. 2020). Shimoji (2014) described the method used to derive operational AMVs from the 10-min (full-disk) and 2.5-min (target) observations of Himawari-8; the method includes the averaging of cross-correlation surfaces obtained from forward and backward template matching.

In September 2020, Meteorological Research Institute and Japan Meteorological Agency (JMA) conducted a special observation of Typhoon Haishen (2020); Fig. 1 shows its track and intensity from best track data (section 2a). During a 3-day period from 3 September 2020, Himawari-8 was operated to observe Haishen at intervals around 30 s. The imaged area is around 1000 km in longitude and 500 km in latitude, the latter being the width of a single scan of the Advance Himawari Imager (AHI) on board Himawari-8 (Okuyama et al. 2018).

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(a) The track and (b) the estimated intensity of Haishen (2020). Gray shading in (b) indicates the analysis period. The track in (a) is based on the JMA best track data, and the intensity in (b) is taken from the JMA (solid) and JTWC (dashed) best track data; black curves show central pressure, and red curves show maximum wind speed, which is meant to approximate 10- and 1-min maximum sustained wind speeds at 10 m above the sea for JMA and JTWC, respectively.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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In this study, we develop a method to obtain AMVs at an unprecedented spatiotemporal resolution by utilizing the high sampling rate. The method employs more than three images by tracking subimages in a Lagrangian manner. A similar but different approach has been used to track clouds in the atmosphere of Venus (Horinouchi et al. 2017, 2020b), in which correlation surfaces with different time intervals are averaged.

The AMVs obtained by our method are used to study the dynamics of the eye of Haishen. As in TH20, we quantify the motion of clouds in the boundary layer of an eye, but unlike TH20, we derive horizontal distributions of velocities, which include asymmetric motions. They are found to substantially include low-wavenumber stationary and transient components. Of particular interest would be a wavenumber-1 feature and its role in the angular momentum transport within the eye. The results are consistent with the theoretical and numerical studies by Nolan and Montgomery (2000, hereinafter NM00) and Nolan et al. (2001), whose importance appears to have been overlooked.

The rest of the paper is organized as follows. The data and methods used are described in section 2. Examples of the present AMVs are presented in section 3 to explain their quality and characteristics including the altitude they represent. Main results are presented in sections 4 and 5, quasi-stationary features are shown in section 4, and transient features and their role to change the quasi-stationary features are shown in section 5, a barotropic simulation to aid interpretation is presented in this section. A discussion to further interpret the transient features is presented in section 6, and conclusions are drawn in section 7.

2. Data and method

a. Data

We used image data from Himawari-8 (Bessho et al. 2016) obtained by a special observation conducted twice a minute at variable time intervals at around 30 s (mostly between 25 and 35 s). Cloud tracking was conducted by using the reflectivity of visible light at 0.64 μm (band 3), whose resolution is 0.5 km at the subsatellite point. The band-3 radiance is scaled to mimic albedo (nondimensional; values up to around 1.2). Infrared brightness temperature at 10.4 (band 13) and 11.2 μm (band 14) was used for the parallax correction and thin-cloud masking described below. Their resolutions are 2 km at the subsatellite point. We also used two independent best track data by Regional Specialized Meteorological Center Tokyo, JMA and by Joint Typhoon Warning Center (JTWC). Environmental data were based on the Japanese 55-year Reanalysis (JRA-55; Kobayashi et al. 2015); we used pressure-level data with a resolution of 1.25°.

The period of data used in the present study is from 2200 UTC 3 September to 0800 UTC 4 September 2020. Corresponding local time is from ∼7 to ∼17 h, so visible reflectivity is available. The 30-s imaging was paused three times for 0240–0250, 0310–0330, and 0440–0500 UTC. The analysis period corresponds to the time just before the intensity was maximized (Fig. 1). The upper clouds in the eye started to diminish on 3 September, and the eye was mostly cleared up at around 0000 UTC 4 September. Figure 2 shows the environmental winds during the analysis period. Its definition is given in the figure caption; the profile is not very sensitive to the area to compute it. The environmental shear defined as the difference between 200 and 850 hPa is 6.1 m s⁻¹ to the clockwise direction of 37° from the east, which is a moderate southeastward shear (zonal and meridional components are 4.9 and −3.7 m s⁻¹, respectively). The translation speed was 4.6 m s⁻¹ to the anticlockwise direction of 145° from the east, which is northwestward and nearly opposite to the shear direction (zonal and meridional components are −3.8 and 2.6 m s⁻¹, respectively).

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(a) Environmental winds of Haishen during the analysis period based on the JRA-55 reanalysis: horizontally and temporally averaged zonal (red) and meridional (black) velocities over ±7° in longitude and latitude with respect to the TC center are obtained at 0000 and 0600 UTC 4 Sep 2020, and the results at the two times are averaged. (b) As in (a), but for the hodograph representation. Numbers show pressure levels (hPa).

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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b. Image resampling

The band-3 visible radiance data are resampled by the following steps:

• Parallax correction (described in appendix A)

• Storm-center refinement (described in appendixes B and C)

• Final sampling with respect to the refined centers

The parallax corrected images are used for the storm-center refinement and the final resampling. We employ the two definitions of the storm center as follows:

1. Eyewall relative center, which is defined in terms of the inner edge of the eyewall clouds at around 5 km in altitude.

2. Circulation relative center, which is defined in terms of the low-level circulation obtained from AMVs obtained with the images gridded using the centers specified by the method 1.

These centers are defined as quantities on the time scales of 1–2 h, and they are identified every hour. Their relative positions are found to be roughly unchanged over the analysis period. The center-finding methods are fully described in appendixes B and C, and the results are shown in Table 1.

Table 1

JMA best track center locations at 0000 and 0600 UTC 4 Sep 2020, and the correction by the method 1 (eyewall relative) and 2 (circulation relative) from 2200 to 0000–0800 UTC 4 Sep 2020. Best track center locations are shown by longitude (λ) and latitude (φ) in degrees, and the corrections are shown for longitude (dλ) and latitude (dφ) in 0.01°.

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Corresponding to the two types of center definitions, we obtain two sets of the coordinate definition for final resampling: the coordinate origin of the one is the eyewall-relative storm center, and the coordinate origin of the other is the low-level circulation center. For each of them, we resample the parallax corrected images by a Cartesian coordinate grid on the equidistant cylindrical projection. There, two coordinates are taken eastward (x) and northward (y), and the grid resolution is set to 500 m.

3. High-frequency, high-resolution AMVs from ultra-rapid-scan images

In this section, we show some examples of our AMVs and describe their characteristics. Also described are the altitudes that they represent and dependency on imaging time intervals. The online supplemental movie shows the tracking results every 3 min by showing the positions of the center of the template subimages according to the backward and forward tracking, from which our AMVs are derived. In this movie, the results before applying the postprocessing in appendix D, section b (the screening with split-window images) are shown. The movie indicates that our tracking generally captures cloud motions well. There are some cases in which the Lagrangian tracks do not appear to represent cloud motions; for example, when the clouds are very streaky, the trajectories sometimes move along the streaks unnaturally. However, these artifacts are expected to be small if the AMVs are averaged over time, as done in our analyses (sections 4 and 5).

Figure 3 shows AMVs at a 1-h interval since 2240 UTC. The figure demonstrates that AMVs are obtained at most of the grid points at the resolution of 2 km, except where reflectivity is nearly featureless. As the contours representing nominal cloud-top altitudes indicate, upper-clouds intruded into the eye. However, the postprocessing based on the split-window technique (appendix D, section b) rejects many of the AMVs obtained where optically thin cirrus clouds are present, as indicated by the white arrows.

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Example of AMVs obtained at the Cartesian grid (vectors) overlaid on the reflectivity at the reference (central) time of tracking (gray shading), (a) 2240, (b) 2340, (c) 0040, and (d) 0140 UTC. The arrow on the lower-right corner of each panel indicates the length corresponding to 50 m s⁻¹. Black arrows are AMVs used in this study, and white arrows are AMVs rejected by the postprocessing in appendix D, section b. Contours shows the cloud-top height obtained from band-13 brightness temperature and the JRA-55 reanalysis (interval: 2 km). The coordinate origin is eyewall relative.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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As in Fig. 3, the nominal cloud-top altitudes are minimized at around 2 km, even where reflectivity is small and more-or-less clear-skied. This is because of the weak but nonnegligible absorption of the 10.4-μm (band 13) IR by water vapor, which is abundant in the boundary layer. The fact that even the cloud-free sea surface is labeled as being at around 2 km indicates that the clouds in the eye may have their tops at altitude lower than the values of the IR- and JRA-based geopotential height. Given the typical thermodynamical structure of developing intense TCs with eyes, the boundary layer is likely capped by thermal inversion, and the clouds in the eye should reside in the boundary layer (except for upper clouds intruded from the eyewall; we did not find hints of hub clouds). The actual boundary layer depth can be as low as a few 100 m near the eye center (e.g., Willoughby 1998), but for Haishen, it is unknown because of the lack of sounding observations.

Observational information on the cloud bottom altitudes is also not available for the present case. However, past studies with dropsondes have indicated that relative humidity is nearly vertically uniform in the boundary layer in the eyes except the bottom ∼100 m (e.g., Zhang et al. 2013; Yamada et al. 2021). Therefore, the boundary layer clouds are likely to present nearly across the boundary layer except near surface. Therefore, the AMVs near the center of the eye, where clouds tend to be separated with each other, are expected to be close to the mean horizontal velocities over the boundary layer except near surface. Except near the center (or where the nominal cloud tops are around 3 km or higher), clouds are packed with each other, so the AMVs are expected to reflect motion of brightness features appearing near the cloud top.

To demonstrate the usefulness of the 30-s observations, AMVs corresponding to 1- and 2-min image sampling are shown in Fig. 4. These are obtained by fixing the initial template subimages throughout the twice-a-minute tracking, so the search areas for horizontal displacement are the same as in the default tracking (∼5 pixels). In other words, the tracking is aided by the 30-s sampling, so erroneous template matches are suppressed as in the default tracking. Even in this case, however, the number of valid AMVs decreases as the total time interval increases. This is because of the deterioration of cross correlation (i.e., cloud-morphology resemblance) over time. This result demonstrates the usefulness of high-frequency sampling and the need for more elaborate methods to conduct cloud tracking with the operational 2.5-min target observations by Himawari-8; an example is TH20.

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As in Fig. 3d, but for showing the results of the sensitivity test regarding the time sampling. Actually, these are the AMVs obtained by using the initial template subimages at the reference time throughout the Lagrangian tracking conducted over (a) 1 min (two steps both in forward and backward) and (b) 2 min (four steps).

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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4. Quasi-stationary features

As will be shown in section 5, the motion in the eye varied with a period around 1 h. Here, we filter it out by taking averages over 2 h to examine more slowly varying components. As introduced in section 1, this study is aimed at investigating motions within the eye. Before doing so, we briefly touch on features around the eyewall in the next paragraph.

As seen in Fig. 5, Haishen had a nearly circular eye. In Fig. 5a (averaged between 2300 UTC 3 September and 0100 UTC 4 September), the geopotential height contours between 5 and 12 km exhibit greater separations in the west-southwest direction than in the other directions (note that the height contours to the south of the eye at the time is affected by inward intrusion of upper clouds in the eye). This observation indicates a tilt of the inner edge of the eyewall clouds to the west-southwest direction, which is to the right of the environmental shear (in the southeast direction). During the analysis period, the tilt direction slightly rotated counterclockwise; in Fig. 5j (averaged between 0500 and 0700 UTC), the height contours between 4 and 12 km exhibit greatest separations to the south, indicating a southward tilt. Also, the eye was contracted during the analysis period. The middle column of the figure (Figs. 5b,e,h,k) shows 2-h mean horizontal winds and their speeds (throughout this paper, wind velocities are storm relative as noted in section 2c). Because of the expected warm core structure and the associated vertical shear, the present 2-hourly horizontal wind speeds on the middle to upper part of the eyewall are likely smaller than those at around the boundary layer top below, where winds are faster. Accordingly, the 2-hourly relative vorticity shown in Figs. 5c, 5f, 5i, and 5l should be viewed with caution; the negative vorticity along the eyewall is likely an artifact arising from eyewall inclination, which was ignored in the vorticity computation.¹ Note that, in an axisymmetric vortex, zero-vorticity along a surface indicates zero-angular momentum gradient along the surface. Thus, the present result is not unexpected from cloud tracking along the inner edge of the eyewall.

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Results averaged over 2 h from (a)–(c) 2300, (d)–(f) 0100, (g)–(i) 0300, and (j)–(l) 0500 UTC. (left) Radiance (gray shading) and band-13-based geopotential height (contours; interval: 1 km). (center) Horizontal wind velocity (vectors) and speed (color shading; shown where valid AMVs are obtained over more than one-third of the period); contours are as in the left panels, but the interval is 2 km. (right) As in the center panels, but for the relative vorticity derived with central differentiation, which corresponds to the vertical component of vorticity if the AMVs are obtained at the same altitude (see the text and the footnote therein for what this quantity, ζ_c, represents when adjacent AMVs are obtained on an inclined surface); the vorticity is smoothed by the 3 × 3 two-dimensional running mean (color shading). The coordinate origin is eyewall relative.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The 2-h-mean wind speeds in Fig. 5 are maximized near the bottom of the inner edge of the eyewall, where the nominal altitude is around 4 km. Their maximum values are between 50 and 60 m s⁻¹ depending on time. In strong TCs with eyes, it is expected that wind speeds are maximized in the eyewall [see, e.g., Kossin et al. 2007 who compared the eye radii and the radii of maximum wind (RMW) observed with aircraft], which is optically unobservable from space. Therefore, the maximum storm-relative wind speed in the entire storm should be greater than 60 m s⁻¹.

Winds in TCs are often characterized with radial and tangential components, but they depend on the selection of coordinate origin. Figure 6 shows the results by using the origin set in terms of the lower (altitude 5–6 km) part of the inner edge of the eyewall (eyewall relative; section 2b). In this coordinate, radial wind (second column) and angular velocity (right) have substantial wavenumber-1 features.

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As in Fig. 5, but for showing quantities that depends on the coordinate origin, which is eyewall relative. Here, 2-hourly means were derived by using AMVs obtained where the band-13 brightness temperature is greater than 260 K. Tangential wind, radial wind, absolute angular momentum, and angular velocity around the coordinate origin are shown from left to right. Contours are geopotential height (interval: 2 km).

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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This result indicates that the center of low-level circulation was not at the coordinate origin. In fact, horizontal wind speeds shown in Fig. 5 (middle column), which do not depend on coordinate origins, are minimized a few kilometers away from the eyewall relative origin. We therefore defined another origin with respect to the low-level circulation (circulation relative; section 2b). The distance between the eyewall relative and circulation relative centers is around 3–5 km (Table 1). Figure 7 shows the results using the circulation relative origin. As expected, the wavenumber-1 components in both radial wind and angular velocity are substantially reduced from those in Fig. 6.

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As in Fig. 6, but for using the circulation relative origin.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The wavenumber-1 radial wind distribution in the conventional eyewall relative coordinate (Fig. 6) is directed inward and outward to the right and left, respectively, of the northwestward translation direction (see section 2a). This result is consistent with that obtained by Kepert (2010) by using a multilayer boundary layer model (see their Fig. 8a), and it is slightly different from Shapiro (1983) who used a slab Ekman layer model. Therefore, the present wavenumber-1 feature can be explained as a boundary layer response to the TC translation. As shown in Fig. 7, the wavenumber-1 component significantly reduces if the coordinate is taken relatively to the circulation center. This suggests that the wavenumber-1 feature is predominantly explained by the shift of the center of boundary layer circulation to the rearward of translation. (See appendix C and Fig. C1 for the cloud-top altitudes.)

Zhang et al. (2013) composited dropsonde results in terms of shear direction, not translation direction. Their Fig. 6 shows the results for radial winds, to which our results should be compared at radii smaller than the RMW. The differences among the four quadrants in their figure are small and varies with radius even within RMW. Thus, the comparison is not straightforward, and it is difficult to draw conclusions.

One might question that the differences between the eyewall-relative and the circulation-relative centers may be because of vortex tilt, since they are evaluated for different altitudes. This might be true to some extent. However, the two definitions are based on different quantities, so to define the tilt from them would be difficult. Moreover, if the differences are dominated by tilt, its direction is roughly northwestward throughout the analysis period; this direction does not agree with the tilt between 4 and 12 km (argued in the second paragraph of this section). Therefore, it is not straightforward to interpret the differences by tilt, if possible.

Time evolution of azimuthal-mean quantities relative to the overall low-level circulation center is shown in Fig. 8 as 10-min running means. As the eye contracted with time, tangential wind in the eye increased at each radius, and the angular velocity became nearly uniform at the end of the analysis period. Note that the results at radii greater than where tangential winds are maximized reflect motions of the eyewall and upper stratiform clouds.

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Azimuthal-mean and 10-min running-mean (a) tangential velocity and (b) angular velocity with respect to the circulation relative origin. The abscissa is date and time in UTC.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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5. Transient features and angular momentum redistribution

a. Traveling wavenumber-1 disturbance as a rotating circulation center

Figure 9 shows 6-min mean horizontal winds every 12 min over an hour. The wind speed in Fig. 9a is minimized to the south to southeast of the eyewall-relative storm center. The flows round the minima are cyclonic, as indicated by the arrows representing horizontal winds. This wind speed minimum rotates counterclockwise to return close to the original position by one hour (Fig. 9f). It orbits around the mean circulation center, as can be seen in Fig. 10. Thus, in this eyewall-relative coordinate, the distance between the wind minimum and the coordinate origin varies with time.

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Horizontal velocity (vectors), speed (color shading), and geopotential height (contours; interval: 2 km) averaged over 6 min, shown every 12 min for an hour from 0100 UTC. Velocities are shown only where at least three valid AMVs are obtained among the six to average. The coordinate origin is eyewall relative.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The time evolution of the location at which the 6-min-mean horizontal wind speeds smoothed by the 5 × 5 horizontal running mean are minimized. The results are shown since 0100 UTC, changing colors every 1 h. The coordinate origin is eyewall relative. The line is broken where the observation was paused.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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One may think that this traveling wind speed minimum can be interpreted as a mesovortex embedded within the eye. However, this region is not accompanied with a distinct vorticity peak (not shown), so it is not appropriate to call it a mesovortex. See section 5d for the relation between the circulation center and vorticity distribution in an idealized simulation.

The radial velocity u at the radius (r) of 8 km, which was defined against the overall low-level circulation center, (Fig. 11a) exhibits a clear wavenumber-1 feature, rotating at a period around 1 h; the rotation speed slightly increased during the analysis period (the rotational period was around 1.1 and 1 h before and after 0100 UTC, respectively). The wavenumber-1 disturbance was obscure before 0000 UTC, which may be partly because a substantial fraction of the eye was covered with upper clouds. The amplitude of the radial wind disturbance was increased with time after 0000 UTC to reach around 10 m s⁻¹ at 0200 UTC. Figure 12 shows the radial wind variance averaged for r ≤ 18 km, which is defined as $⟨r\overline{{u^{\prime }}^2}⟩/⟨r⟩$, where overbar, prime, and angle brackets represent azimuthal mean, deviation from azimuthal mean, and radial mean, respectively. Here, the factor of r is applied to incorporate the areal factor. The variance increases from ∼10 m² s⁻² at 2300 UTC to ∼50 m² s⁻² at 0500 UTC. The wavenumber-1 feature in the radial velocity is confined in the eye, being small at r greater than 20 km (Fig. 11b). Its phase is nearly independent of r.

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(a) 10-min running-mean radial wind u at the radius of 8 km with respect to the low-level circulation (circulation relative coordinate). Values are shown where AMVs are obtained originally (before running mean). The ordinate is the azimuth θ (counterclockwise from the east). (b) As in (a), but for the time–radius section of the wavenumber-1 component at θ = 0.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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Radius-weighted-mean of the radial wind variance over 0 < r ≤ 18 km, $\hspace{0.17em}⟨r\overline{{u^{\prime }}^2}⟩/⟨r⟩$, subject to a 5-min running-mean (in the circulation relative coordinate).

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The orbital angular velocity of ∼1.7 × 10⁻³ s⁻¹ (note: periods of 1 and 1.1 h correspond to the angular velocities of 1.75 × 10⁻³ and 1.59 × 10⁻³ s⁻¹, respectively) is greater than the rotational angular speed of the tangential wind at r < 15 km before 0300 UTC (Fig. 8b). The difference suggests that the wind speed minimum propagated as a wave with respect to the local flow, which supports the earlier notion that it should not be regarded as a mesovortex.

b. Angular momentum variation and transfer

As shown in section 4, the rotation angular velocity of Haishen’s eye sped up during the analysis period. Here, we further examine it in terms of the periodic variability shown above. The following equation can be derived from the azimuthal component of the momentum equation:

$\frac{d}{dt}{{\displaystyle \int }}_0^{r}r\overline{m}dr=-r\overline{mu}-{{\displaystyle \int }}_0^{r}r\frac{\partial (\overline{mw})}{\partial z}dr+{{\displaystyle \int }}_0^{r}r\overline{X}dr,$(1)

where t is time, m is the absolute angular momentum per unit mass [m = rυ + (fr²/2), υ is tangential wind, and f is the Coriolis parameter], z is altitude, and X is the frictional force in the momentum equation. Note that, if the Coriolis term is small, the integrand $r\overline{m}$ is nearly proportional to $r^3\overline{\omega }$, where $\overline{\omega }$ is the angular velocity at r. Therefore, the radially integrated angular momentum over radius from 0 to r, $M(r)\equiv {\int }_0^{r}r\overline{m}dr$, is dominated by the rotation at around r.

We show the time evolutions of M(r) for the eyewall relative and circulation relative coordinates in Figs. 13a and 14a, respectively, at r = 16 km. It oscillates at a period around 1 h in the former (Fig. 13a). This is a direct manifestation of the transient wavnumber-1 disturbance shown in section 5a, which is understood from the Kelvin circulation theorem; if the region of low vorticity changes its distance to the coordinate origin, mean tangential winds at a radius change accordingly. As expected, therefore, the periodic (at a period of ∼1 h) fluctuation of M(r) in the circulation relative coordinate (Fig. 14a) is much weaker.

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Time evolutions of radially integrated angular momentum up to r = 16 km and its forcing in the eyewall relative coordinate. (a) M(r) smoothed with the running mean over 11 min. (b) (d/dt)M(r) (derived after applying the running mean; black), and the 11-min running means of the horizontal eddy forcing $-r\overline{mu}$ associated with the wavenumber-1 components of m and u (red) and that associated with higher wavenumbers (blue). All the quantities were derived after interpolating data missing along azimuth to avoid possible biases in azimuthal means.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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As in Fig. 13, but for using the circulation relative coordinate.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The time derivative of M(r) in the eyewall relative coordinate (black curve in Fig. 13b) is almost entirely explained by the wavenumber-1 horizontal eddy forcing (red curve). This is what it should be, since the oscillation of M(r) in this coordinate is because of the offset of the coordinate origin from the center of the periodical vortex rotation; in other words, this is a matter of consistency rather than a causality. What this result essentially indicates is that our cloud tracking results are good enough to capture the relation faithfully.

The integrated angular momentum M(r) increases over the analysis period, as seen both in Figs. 13a and 14a. In the circulation relative coordinate, (d/dt)M(r) is not closely followed by the wavenumber-1 forcing, but it tends to be positive (Fig. 14b). Figure 15 shows their time averages at various radii. They are both positive and have similar magnitudes. This result indicates that the overall speed up of the low-level rotation in the eye can be explained by the angular momentum transport by the wavenumber-1 disturbance, which is essentially a wobble of vorticity distribution in the eye. Note that we have not quantified the second and third terms on the right-hand side of Eq. (1), nor the wavenumber-0 component of the first term. Therefore, the quantitative agreement in Fig. 15 might be a coincidence. However, the wavenumber-1 forcing is consistent with earlier theoretical studies, as argued in what follows.

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(d/dt)M(r) (black curve) and the wavenumber-1 horizontal eddy forcing (red curve) averaged over 2300–0700 UTC, shown as functions of r in the circulation relative coordinate.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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d. Barotropic simulation

To demonstrate the algebraically growing instability, the nondimensional barotropic model (section 2d) was run with the initial vorticity distribution (top-left panel of Fig. 16):

$\zeta =a_1\hspace{0.17em}\exp \left[-\frac{x^2+y^2}{2{\sigma }_1^2}\right]-a_2\hspace{0.17em}\exp \left[-\frac{{(x-x_2)}^2+y^2}{2{\sigma }_2^2}\right]+b,$(2)

where σ₁ = 0.1, σ₂ = 0.05, a₁ = 0.7, a₂ = 0.525, and x₂ = 0.05; b is set to make the areal mean vorticity zero. The subtraction of the two Gaussian distributions yields a smooth vortex ring, but ζ is nonzero at r = 0 as in the observation because a₁ > a₂. The nonzero introduces x₂ asymmetry to induce unstable motions. The vorticity distribution (2) is meant to crudely express inner-core vorticity distribution with weak asymmetry. NM00 used nearly stepwise ζ distributions with nonzero ζ at r = 0.

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Barotropic simulation result: vorticity (color shading), streamfunction (contours), and the position of its minimum (+ marks) are shown at an interval of 20 nondimensional times.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The simulation results are visualized in Figs. 16 and 17. As shown in NM00, the vorticity minimum rotates cyclonically, and its distance to the coordinate origin (which is fixed) increases with time (color shading in Fig. 16). However, the distance of the location where horizontal velocity is minimized (+ marks in Fig. 16), which is referred to as the circulation center, does not increase much. Note that the positions of the vorticity minimum and the circulation center are situated oppositely with the coordinate origin. This is because the circulation center is associated with a cyclonic anomaly, and the vorticity minimum is an anticyclonic anomaly. As can be computed from Fig. 17, both the period of the disturbance and the maximum azimuthal-mean rotation period are ∼32 nondimensional time. This agreement is consistent with the algebraically growing instability theory. Also, the rotational angular velocity increases with time within RMΩ. This is because the disturbance transports angular momentum inward. The central pressure decreases in this process, but the maximum rotation angular velocity does not increase within the period of simulation shown in the figure (for reference, it eventually increases near the center; if the simulation is extended, because the low vorticity region is ejected as shown by Nolan et al. 2001).

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Barotropic simulation result: azimuthal-mean angular velocity (color shading) and radial velocity at θ = 0 (contours; solid and dashed for outward and inward, respectively). The abscissa and ordinate are the nondimensional time and radius, respectively.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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The observational features of the transient wavenumber-1 disturbance and the homogenization of the rotational angular velocity shown above are consistent with the present simulation. However, the instability does not explain the observed contraction of the eye. Also, it does not explain that the maximum angular velocity was increased with time (Fig. 8b). We suppose that these disagreements were caused by other factors that can be understood in an axisymmetric framework (i.e., through the secondary circulation).

6. Discussion: Nature and role of the wavenumber-1 instability

In this section, we discuss the nature of the algebraically growing wavenumber-1 instability to demonstrate their possible ubiquity and roles. Here we begin by interpreting some basic features of the instability as a discussion basis. It leads to the suggestion that this instability is an important process for translating TCs and TCs in sheared environment, in which vorticity generation tends to be asymmetric.

The term I can be interpreted as radially integrated wavenumber-1 asymmetry in vorticity. The nature that the growth speed is proportional to I indicates that the instability is faster if this asymmetry is greater. This is understood in such a way that the vorticity hollow ejection is easier if the initial vorticity ring is thicker or “heavier” on one side of the ring and thinner or “lighter” on the other side.

In the real atmosphere, TCs in sheared environment tends to have greater low-level inflow from the down-shear directions, which tends to create wavenumber-1 asymmetry in the eyewall clouds and vorticity (e.g., Braun and Wu 2007). Also, translating TCs tend to have wavenumber-1 asymmetry as have been introduced. The discussion presented so far in this section further suggests that such wavenumber-1 asymmetries are unstable, and that it can promote ejection of low vorticity in the eye through the wavenumber-1 instability, which acts to accelerate the mean tangential flow near the center.

In our simulation results shown in Fig. 16, it is evident that nonlinear processes evolve with time, in which shear elongates the arc of vorticity peak. Figure 18 shows its further time evolution. It is highly nonlinear; the vorticity hollow becomes smaller with time by erosion following partial filamentation (similarly to the textbook examples for Kirchhoff’s elliptic vortices), and it is ejected by t = 1600. This ejection process is remarkably similar to the “slow monopole” type of the typical longtime behaviors of barotropically unstable vortices studied by Hendricks et al. (2009) (their Fig. 7 and Table 6). This behavior occurs when the initial instability at wavenumbers greater than 1 is matured but still a low-vorticity region is left near the center; a vorticity monopole is formed after ejecting it. Therefore, at nonlinear phase, such ejection appears a major driver of the mean tangential flow acceleration near the TC center. In Hendricks et al. (2009), the vortex that do not have exponentially growing modes did not collapse within a long but finite time that they specified. This is likely because the initial disturbance that they added was small, which is not necessarily realistic for TCs as argued above. At finite amplitude, they can change to the “slow monopole” behavior. As such, the nonlinear vortex hollow ejection appears one of the major processes that explains the longtime behavior of low-level vorticity distribution in TC eyes. The algebraically growing waenumber-1 instability appears a good model for that.

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As in Fig. 16, but for later times: t = 400, 800, 1200, and 1600.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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It is noteworthy that filamentary features of vorticity seen in Fig. 18 resemble cloud streets that frequently appear in visible cloud imagery (e.g., Kossin 2002). However, our AMVs do not appear to be accurate enough to diagnose instantaneous small-scale vorticity distribution.

7. Conclusions

We developed a method to derive AMVs from high-frequency imaging, in which template matching is conducted for multiple steps both forward and backward in a Lagrangian manner. The method was applied to the 30-s special observation of Typhoon Haishen (2020) with the Himawari-8 satellite. Various screening was applied to secure quality. The analysis period is from 2200 UTC 3 September to 0800 UTC 4 September 2020, which is a late phase of Haishen’s intensification. We derived storm-relative AMVs from visible cloud images at 0.64 μm (band 3). Near the eye center, the AMVs are likely to represent horizontal flows in the boundary layer.

The overall low-level circulation center, which was obtained by filtering out periodic transient variability, was situated several kilometers away from the TC center determined in terms of the lower part of the eyewall’s inner edge. The direction of the shift is to the rearward of the storm translation as schematically illustrated in Fig. 19 with the cross mark. This shift can be explained as being caused by surface friction, as suggested by a boundary layer simulation of Kepert (2010). Over the analysis period of 10 h, azimuthal-mean tangential wind around this center was increased at each radius within the eye, and the rotational angular velocity was nearly homogenized.

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Schematic illustration of the stationary (on the time scale of ∼1 h) and transient dynamical features found in the eye of Haishen.

Citation: Monthly Weather Review 151, 1; 10.1175/MWR-D-22-0179.1

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A transient periodic variability of wind distribution was found to persist in the eye; the instantaneous circulation center as a local minimum of horizontal wind, around which flows are cyclonic, orbited at a period of ∼1 h around the overall low-level circulation (as schematically illustrated in Fig. 19 by curved arrows). The orbital angular speed of this disturbance was close the maximum angular speed of azimuthal-mean tangential winds. A similar periodic feature was reported by Marks et al. (2008) by deducing pressure distribution from an aircraft observation. This study further elucidated its dynamical impact. This wind disturbance was found to transport angular momentum inward within the eye. Its effect was quantitatively comparable to the increase in the mean tangential winds.

These features and effects of the periodic wind disturbance are consistent with the algebraically growing wavenumber-1 instability studied theoretically and numerically by NM00 and Nolan et al. (2001). Among the four features shown near the end of section 5c (itemized), the first one regarding the orbital angular velocity is especially useful to judge the consistency with this instability; the disturbance travels near the rotational angular speed at RMΩ, which is faster than the local rotational angular speed before it is homogenized. Our numerical simulation further demonstrated the consistency. The instability, on the other hand, does not explain the observed eye contraction and the maximum angular speed increase with time as seen in Fig. 8. These features are likely caused by processes that can be explained in the axisymmetric framework.

The algebraically growing wavenumber-1 instability acts to slowly enhance asymmetry in ring-shaped vorticity distribution, moving the vorticity minimum away from the center to eventually eject it (after some longtime). This behavior is common to the slow formation of vorticity monopole, which occurs after an initial exponential barotropic instability at wavenumbers greater than 1 is matured, as studied by Hendricks et al. (2009). This ejection process is likely understood in the framework of the algebraically growing wavenumber-1 instability. Since the translation by mean flow and environmental shear tend to induce wavenumber-1 asymmetry in the vortex ring of the inner core of TCs, the instability can occur without initial barotropic instability at high wavenumbers. Therefore, the slow wavenumber-1 instability might be one of the major processes that redistribute potential vorticity in the eye to increase rotation near the center. Further studies would be needed to elucidate its prevalence and its importance in the dynamics of TCs.

Data availability statement.

Himawari-8 satellite data at 2.5-min time resolution are publicly available from many data providers (e.g., Chiba University http://www.cr.chiba-u.jp/databases/GEO/H8_9/FD/index.html). Due to its proprietary nature, the 30-s Himawari-8 data cannot be made openly available. The availability of the 30-s observation data is subject to formal collaboration with Meteorological Research Institute. The best track data of JMA and JTWC are publicly available from https://www.jma.go.jp/jma/jma-eng/jma-center/rsmc-hp-pub-eg/besttrack.html and https://www.metoc.navy.mil/jtwc/jtwc.html?best-tracks, respectively.