1310 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58

Topographic Effects on Barotropic Vortex Motion: No Mean Flow

HUNG-CHI KUO Department of Atmospheric Science, National University, , Taiwan

R. T. WILLIAMS Department of Meteorology, Naval Postgraduate School, Monterey, California

JEN-HER CHEN AND YI-LIANG CHEN Central Weather Bureau, Taipei, Taiwan

(Manuscript received 19 July 1999, in ®nal form 23 October 2000)

ABSTRACT The impact of the island topographic ␤ effect on hurricane-like vortex tracks is studied. Both f plane and spherical geometry without a mean ¯ow are considered. The simulations used in this study indicate the existence of a track mode in which vortices are trapped by the topography and follow a clockwise island-circulating path. The trapping of a hurricane-like vortex can be interpreted in terms of the in¯uence of the island topographic ␤ effect on the vortex track. Experiments on the f plane indicate that the drift speed along the clockwise path is

proportional to the square root of ␤evmax. The applicability of the square root law on the f plane is dependent

on the degree to which the local ␤e effect is felt by the vortex. The experiments on the sphere also demonstrate

that the speed along the clockwise path is larger for a vortex with a larger maximum wind vmax. The occurrence

of hurricane-like vortex trapping, however, is not sensitive to the value of vmax. When there is no background ¯ow, the vortex will drift to the northwest in the presence of the planetary vorticity gradient. The ␤ drift speed

acts to keep the vortex from being trapped. The insensitivity of the vortex trapping to vmax on the sphere appears to be due to the possible cancellation of stronger planetary ␤ and topographic ␤ effects. The experiments suggest that the topographic scale must be comparable to (if not larger than) the vortex radius of maximum wind for the trapping to occur. Nonlinear effects are important in that they hold the vortex together and keep it moving without strong dispersion in the island-circulating path. This vortex coherency can be explained with the ␤ Rossby number dynamics. The global shallow-water model calculations used in this study indicate that the vortex trapping increases with peak height, topographic length scale, and latitude (larger topographic ␤ effect). In general, the trapping and clockwise circulating path in the presence of a planetary vorticity gradient will occur if the scale of the topography is greater than the vortex radius of maximum wind and if the planetary ␤ parameter is less than the topographic ␤ parameter.

1. Introduction 1998), the vertical wind shear over the (e.g., DeMaria 1996), and the oceanic temperature on The problem of tropical cyclone forecasting is some- the hurricane intensity change (e.g., Bender et al. times divided into two parts: intensity prediction and track prediction. At present, we seem to have little skill 1993). Thus, accurate intensity prediction requires a in predicting tropical cyclone intensity changes. The- realistic interaction of moist physics and dynamics in oretical work (e.g., Hack and Schubert 1986) indicates a numerical model. Moreover, accurate treatment of that intensity changes are related to the relative spatial air±sea interaction in a coupled model may be required distributions of cumulus convection and inertial sta- for intensity prediction. Improved observational and bility within the tropical cyclone. Others emphasize data assimilation capability over the ocean as well as the importance of the approach of an upper-tropo- improved parameterization of the atmospheric bound- spheric synoptic scale trough (e.g., Molinari et al. ary layer will help the cause. The problem of track prediction, which is of consid- erable operational importance, is governed by an ap- parently complex interaction between the large-scale Corresponding author address: Dr. R. T. Williams, Dept. of Me- circulation in which the vortex is embedded and the teorology, Naval Postgraduate School, 589 Dyer Road, Room 254, Monterey, CA 93943. vortex circulation itself. The dominant concept in the E-mail: [email protected] theory and forecasting of tropical cyclone motion is

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``steering'' by the environmental ¯ow. In other words, clockwise. Some important questions regarding a sec- typhoon track prediction can be viewed as a problem ondary low were not answered from the observational of accurately advecting a sharp peak in the potential work. This is because Wang and Shieh's works involved vorticity ®eld by a large-scale environmental back- subjective interpretations in which a mesoscale low- ground ¯ow. In addition to the background steering pressure center was often inferred from wind analysis ¯ow, asymmetric circulations may result from the in- based on the geostrophic relationship. This relationship teraction of the vortex with the background absolute may not be valid due to the presence of topography and vorticity gradient. When there is no background ¯ow, the small spatial scales. The structure changes, and the the vortex will drift to the northwest due to the inter- effect of the secondary low formation have recently action of the vortex with the planetary vorticity gradient been addressed by Chang et al. (1993) using empirical (Adem 1956; McWilliams and Flierl 1979; Chan and orthogonal functions. They found that the spatial scale Williams 1987). This is the so-called ␤-gyre effect (Fior- of the secondary low is signi®cantly smaller than the ino and Elsberry 1989), where an asymmetric circula- scale of the low pressure that can be produced by an tion is induced that advects the vortex. Carnevale et al. interaction of the mean steering ¯ow and the topogra- (1991) studied the propagation of barotropic vortex in phy. a rotating tank and with a numerical model. They found These terrain-induced structure changes, track dis- that cyclones would spiral toward the top of the hill in continuities, and secondary low or vortex formations an anticyclonic sense. This clockwise hill-circulating are also evident in previous numerical simulations. The mode is argued to be similar to the ␤-gyre dynamics, baroclinic modeling studies of Chang (1982) suggest which can be deduced from the rule of ``local north- that secondary vortex centers are formed on the lee side, west'' propagation of cyclones relative to contours of as the vortex is blocked upstream by the mountain range. topography. There are also baroclinic effects that will Lin et al. (1999) argued that the sudden increase of affect the tropical cyclone motion. The baroclinic mod- vorticity over the lee slope was associated with the gen- els of Shapiro (1992) and Flatau et al. (1994) strongly eration of potential vorticity through the mechanisms of indicate that the motion of the tropical cyclone is gov- upstream blocking and wave overturning. Bender et al. erned by the dynamics of the lower cyclonic circulation. (1987) compared simulations of tropical cyclones near Namely, the cyclone moves relative to the background islands in the Caribbean Sea, Taiwan, and in the potential vorticity gradient as though it were a baro- . The maximum height is about 4300 m in tropic vortex with a vertically averaged wind pro®le. Taiwan and 1500 m for the Philippines. Only in the The diabatic processes also increase the vertical cou- Taiwan simulation was a secondary low formed; how- pling. Thus, the ®rst-order propagation effect in baro- ever, the secondary low center was simulated to occur clinic conditions may be similar to the barotropic fore- much closer to the mountain peak as compared to the cast. observational study of Chang et al. (1993). The Bender Brand and Blelloch (1974) and Ishijima and Estoque et al. high-resolution simulations produced more real- (1987) studied the orographic effects on a westbound istic intensity changes during typhoon approach, land- typhoon crossing Taiwan. Wang (1980) and Shieh et al. fall, and departure from Taiwan. They also found that (1998) carried out the most comprehensive study of the a region of dry air, which is due to the descending ¯ow Taiwan topographic effect on typhoon movement. After over land, is entrained into the cyclone and is respon- examining the behavior of more than 200 typhoons that sible for the ®lling of the system before landfall. Thus, threatened Taiwan between 1897 and 1996, Shieh et al. their results suggest that a slow-moving typhoon may (1998) showed ®fteen typical tracks for typhoons that start weakening several hours prior to landfall. Yeh and crossed Taiwan. Figure 1 shows 9 of the 15 typhoon Elsberry (1993a,b) studied the upstream de¯ections as tracks from Shieh et al. (1998). The tracks are deter- well as the continuous and discontinuous tracks for mined mainly from the surface data available in Taiwan. west-moving tropical cyclones passing Taiwan. The ef- Interesting cases of discontinuous tracks (e.g., Typhoon fect of the impinging angle to the island was also in- Iris 1955; Typhoon B108 1925; Typhoon Kate 1962), vestigated by Yeh and Elsberry (1993b). They found of secondary lows on the lee (e.g., Typhoon Iris 1955; that in agreement with the observation (e.g., Wang 1980; Typhoon Gilda 1962), and of continuous tracks (e.g., Shieh et al. 1998) vortices tend to track continuously Typhoon Joan 1959; Typhoon Shirley 1960) are present. around the northern end of the island. Discontinuous The thick dashed line for (1989) is the tracks dominate for vortices approaching the central and track for a strong secondary low that was caused by the southern portion of the Taiwan topography. However, chinook wind on the eastern side of central mountain the more intense and rapidly moving vortices moved range. It is obvious from these historical cases (and directly cross over the barrier. Smith and Smith (1995) many more similar recent cases), that the topography of used a 2-km resolution shallow water model to study Taiwan can affect the track of a typhoon. It is also the wake formation on the lee side of an island mountain interesting to see tracks parallel to the topography con- by a drifting vortex. No Coriolis force was included in tours in some cases. Typhoons Bess and Sarah have their experiments. They explored wake formation in clockwise paths, and Typhoon Kate's path is counter- terms of the Froude number and the nondimensional

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FIG. 1. Typical typhoon tracks across Taiwan that are determined mainly from the surface data available in Taiwan. The tracks are (a) Typhoon Joan 1959, (b) Typhoon Irma 1949, (c) Typhoon Shirley 1960, (d) Typhoon Iris 1955, (e) Typhoon B108 1925, (f) Typhoon Kate 1962, (g) Typhoon B78 1917, (h) Typhoon Bess 1952, and (i) Typhoon Sarah 1989. The symbol ``L'' indicate the general position of the lee secondary lows. The thick dashed line for Typhoon Sarah (1989) is the track for a strong secondary low that was caused by the chinook wind on the eastern side of central mountain range. The topographic contour of Taiwan is 500 m.

number M, where M is the ratio of mountain height to the formation of secondary lows, that need to be in- the shallow water depth. Zehnder (1993) studied the vestigated. One may also wonder if the tracks that are in¯uence of large-scale topography such as the Sierra parallel to the topographic contours in the clockwise Madre on barotropic vortex motion with an equatorial sense are due to the topographic ␤ effect when the steer- ␤-plane channel model. He found that vortices were ing ¯ow is weak, as proposed by Carnevale et al. (1991). de¯ected toward the imposed gradient of potential vor- In this paper, we study one aspect of the basic topo- ticity that resulted from the large-scale topography. graphic dynamics and the combination of the local to- Zehnder also investigated track de¯ections as a function pographic ␤ effect and the global ␤ effect on the hur- of vortex distance from the topography of the Sierra ricane-like vortex tracks and on the coherency of the Madre. vortices when there is no mean ¯ow. Both f-plane ge- The in¯uence of topography on the tropical cyclone ometry and spherical geometry calculations are per- strength vortices is very complicated. There are impor- formed. Our simulations suggest the existence of a track tant issues, such as continuous tracks, discontinuous mode in which hurricane-like vortices are trapped by tracks, splitting of upper- and lower-level centers, and the island and follow a clockwise island-circulating

Unauthenticated | Downloaded 09/25/21 11:47 AM UTC 15 MAY 2001 KUO ET AL. 1313 path. Section 2 describes the solution method. The nu- for the numerical experiments. The equivalent grid spac- merical results are presented in section 3, and the con- ings for T96 and T192 resolutions are about 120 and cluding remarks are given in section 4. 60 km, respectively. A square domain of 3000 by 3000 km with a grid spacing of 50 km is used for the f-plane experiments. The numerical method is based on the Fou- 2. Solution methods rier spectral method in space and the fourth-order Run- We use both the double periodic f-plane shallow- ge±Kutta method in time. Following Smith and Smith water model and the global shallow-water model in the (1995), H is set to 5000 m in both models. All the study. The shallow-water model can be written as simulations are without the numerical diffusion added to the right-hand side of (2.1). The idealized island to- ␨ץ ϭ ␣(ϪG, ϪF), (2.1a) pography is given by the function tץ x Ϫ xy22Ϫ y (h ϭ h exp ϪϪ00, (2.6 ␦ץ ϭ ␣(F, ϪG) Ϫٌ2(I ϩ ghЈ), (2.1b) s max ab t []΂΃΂΃00ץ where (x , y ) is the location of the topographic center, hЈ 0 0ץ ϭ ␣(Ϫu*(hЈϪh ), Ϫ␷*(hЈϪh )) Ϫ H␦. (2.1c) a 0 and b 0 the e-folding widths in the two directions, and t ssץ hmax the maximum height of the topography. The char- For the global shallow-water model we de®ne the acteristic topographic slope is determined by the con- operator ␣ as stants hmax, a 0, and b 0. The topographic ␤ effect is re- lated only to the slope of the topography. We employ Bץ Aץ 1 ␣(A, B) ϭϩ, (2.2) a 0/b 0 ratios of ⅓ or ⅔ for most of our simulations. With ␮ these ratios, the orientation of the island topography isץ ␭ץ (Ϫ ␮2 1) and the nonlinear terms as similar to the orientation of the island of Taiwan. The only difference between the circular topography (a 0 ϭ G ϭ U(␨ ϩ f ), (2.3a) b 0) and elliptical topography is that the slope in circular F ϭ V(␨ ϩ f ), and (2.3b) topography is isotropic. The topographic slope is small- er along the major axis in the elliptical topography. U 222ϩ Va The cyclone vortex used in this study has the tan- I ϭ . (2.3c) 2 cos2␾ gential wind v(r) pro®le For the f-plane shallow-water model we de®ne the r 1 r b ␷(r) ϭ ␷ exp 1 Ϫ , (2.7a) operator ␣ as max rbr ΂΃max[] ΂ ΂΃΃ max Bץ Aץ ␣(A, B) ϭϩ, (2.4) and the corresponding vorticity pro®le yץ xץ 2␷ 1 r bb1 r and the nonlinear terms as ␨(r) ϭ max 1 Ϫ exp 1 Ϫ , r 2 rbr max[][]΂΃ max ΂ ΂΃΃ max G ϭ u(␨ ϩ f ), (2.5a) 0 (2.7b)

F ϭ ␷(␨ ϩ f0), and (2.5b) where r is the distance to the vortex center, vmax the u22ϩ ␷ value of v(r) at the radius of maximum wind rmax, and I ϭ . (2.5c) b is a factor that determines the shape of the vortex. 2 Equation (2.7a) has been used previously in many trop- Here, u and v are the zonal and meridional compo- ical cyclone studies (e.g., DeMaria and Chan 1984). We nents of velocity, a is the radius of the earth, ␭ is the follow Chan and Williams (1987) by taking b to be 1.0. longitude, ␾ is the latitude, ␮ ϭ sin␾ is the sine of the Once the initial v(r) and ␨(r) are speci®ed, the initial latitude, U ϭ u cos␾/a, V ϭ v cos␾/a, ␨ is the relative ¯uid depth h is computed from the nonlinear balance vorticity, ␦ is the divergence, h is the ¯uid depth, H is equation. The initial center of the vortex in the f plane the mean equivalent depth, hЈϭh Ϫ H, hs is the to- experiments is near the topography. For the calculations pography, f is the Coriolis parameter, and f 0 is the with the global shallow-water model, we have designed Coriolis parameter for the f plane. In addition, in Eq. the experiments so that the vortex impinges on the to- (2.1c), we have u* ϭ U and v* ϭ v for the global pography near the southern tip of the island from the shallow-water model and u* ϭ u and v* ϭ v for the southeast of the island. Since the ␤ drift of the vortex f-plane shallow-water model. without a mean ¯ow is northwestward, the impinging The numerical method used in the global model in- angles of the vortex to the island are not very different tegration is based on the spherical harmonic spectral in our experiments. Table 1 and Table 2 summarize the method in space and the fourth-order Runge±Kutta relevant parameters for the f plane experiments and method in time. Both T96 and T192 resolutions are used spherical experiments, respectively.

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TABLE 1. f-plane experiments. barotropic vortices over topography in a rotating tank. They reported that a small-scale cyclonic vortex on a a0 b0 hmax Vmax rmax Figure (km) (km) (m) (m sϪ1) (km) relatively broad hill tends to climb toward the top in an 2 200 300 3500 20 100 anticyclonic spiral around the peak. The idea is that a 10 when the equivalent potential vorticity is positive, as b 20 with the hill, the vortex Rossby waves propagate to the 3 400 600 3500 100 c 30 northwest of the local potential vorticity gradient (Adem d 40 1956; McWilliams and Flierl 1979; Chan and Williams a 1500 b 2500 1987; among others). This ``local northwest'' propa- 4 400 600 20 100 c 3500 gation rule implies that a cyclone will spiral toward the d 4500 top of the hill in an anticyclonic sense. The anticylonic a 100 spiral, however, cannot apply when the scale of the vor- b 200 5 200 300 3500 20 c 300 tex is much larger than that of the hill. In laboratory d 400 experiments performed at National Taiwan University, Chu et al. (1998, Fig. 7) also reported Rankine-like vortices orbiting a circular hill in a clockwise direction 3. Numerical results in a rotating tank. Figure 3 shows the sensitivity of the track to the a. f-plane experiments vortex intensity vmax for the f-plane experiments. The

Figure 2 shows the potential vorticity and wind vec- topographic parameters are a 0 ϭ 400 km, b 0 ϭ 600 km, tors at hour 20, 40, 60, and 80 on an f plane. The island and hmax ϭ 3500 m. The vortex scale parameter is rmax Ϫ1 topography parameters are hmax ϭ 3500 m, a 0 ϭ 200 ϭ 100 km, and vmax is 10, 20, 30, and 40 m s in (a), km, and b 0 ϭ 300 km. The vortex parameters are vmax (b), (c), and (d), respectively. The black dot represents Ϫ1 ϭ 20 m s and rmax ϭ 100 km. The normalized potential the center position (potential vorticity maximum) of the vorticity [(H/f 0)(␨ ϩ f 0)/h] contours start at 0 and in- vortex. The vortex initial position is indicated by the crement by 5. The maximum vector is 20 m sϪ1. The ``ϫ'' symbol. The time interval between adjacent black topography contours in all the f plane experiments start dots is 12 h. The total integration time is 120 h. Figure at 1000 m and increment by 1000 m. Figure 2 illustrates 3 demonstrates the insensitivity of the trapping of the an island-circulating vortex track that is very similar to hurricane-like vortices to vmax. Figure 4 is similar to Fig. the water tank experiment reported by Carnevale et al. 3 and shows the sensitivity of the track to the hmax pa-

(1991). Carnevale et al. (1991) studied propagation of rameter. The topographic parameters are a 0 ϭ 400 km

TABLE 2. Sphere-plane experiments.

Ϫ1 Figure a0 (km) b0 (km) hmax (m) Vmax (m s ) rmax (km) ␾0 a 100 4500 b 50 4500 6 150 20 100 45 c 100 2500 d 50 2500 a 600 b 500 7 600 2500 20 100 45 c 400 d 300 a 15 b 25 8 300 600 2500 20 300 c 35 d 45 a 10 b 20 9 300 600 2500 300 25 c 30 d 40 a 1000 b 2000 10 300 600 20 300 25 c 3000 d 4500 a 300 b 400 11 300 600 2500 20 25 c 500 d 600 20 300 12 (initial position of vortex is 300 600 2500 40 300 25 3Њ north) 20 600

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FIG. 2. Potential vorticity and wind vectors at hour 20, 40, 60, and 80 with 50-km grid spacing on a 15Њ north latitude f-plane experiment. Ϫ1 The island topography parameters are hmax ϭ 3500 m, a0 ϭ 200 km, and b0 ϭ 300 km. The vortex parameters are vmax ϭ 20ms and rmax ϭ 100 km. The normalized potential vorticity contours start at 0 and increment by 5. The maximum vector is 22.2, 26.3, 24.4, and 22 m sϪ1 for (a), (b), (c), and (d), respectively. The topography contours start at 1000 m and increment by 1000 m.

and b 0 ϭ 600 km. The vortex parameters are rmax ϭ mum wind of the vortex has to cross the slope of the Ϫ1 100 km and vmax ϭ 20 m s . The value of hmax is 1500, topography for the topographic ␤ effect to be important. 2500, 3500, and 4500 m in (a), (b), (c), and (d), re- Although the trapping of the hurricane-like vortices spectively. Figure 4 indicates that vortex trapping is is insensitive to vmax, the drift speed along the clockwise signi®cant when hmax is increased. Figure 5 is similar path is larger for the vortex with a larger maximum to Fig. 3 and shows the sensitivity of the track to the wind. The speed estimated from Fig. 3 is approximately Ϫ1 vortex scale rmax. The topographic parameters are a 0 ϭ 6.2, 9.9, 12.4, and 16.1 m s in (a), (b), (c), and (d),

200 km, b 0 ϭ 300 km, and hmax ϭ 3500 m. The vortex respectively. It appears that the drift speed is propor- Ϫ1 parameter is vmax ϭ 20ms . The value of rmax is 100, tional to the square root of vmax. This is in agreement 200, 300, and 400 km in (a), (b), (c), and (d), respec- with the scaling law for nondivergent tropical cyclone tively. In the case of Fig. 5d, we have rmax Ͼ a 0, and motion on the ␤ plane (Smith 1993; Smith et al. 1997). the trapping is not as signi®cant as in Fig. 5a. In agree- The drift speed in Fig. 4 is estimated to be 5.4, 6.2, 9.8, ment with Carnevale et al. (1991), Fig. 5 implies that and 12.5 m sϪ1 in (a), (b), (c), and (d), respectively. The the topographic scale needs to be larger than the vortex drift speed in Fig. 4 is proportional to ͙␤e, which radius of maximum wind for the trapping to be signif- shows general agreement with the Smith scaling law. icant. The scale for the trapping means that the maxi- The topographic ␤ effect is de®ned as

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FIG. 3. The sensitivity of the track with respect to the vortex intensity vmax for the f plane

experiments. The topographic parameters are a0 ϭ 400 km, b0 ϭ 600 km, and hmax ϭ 3500 Ϫ1 m. The vortex scale parameter is rmax ϭ 100 km, and vmax is 10, 20, 30, and 40 m s in (a), (b), (c), and (d), respectively. The black dot represents the center position (potential vorticity maximum) of the vortex. The vortex initial position is indicated by the ``ϫ'' symbol. The topography contours start at 1000 m and increment by 1000 m. The time interval between adjacent black dots is 12 h. The total integration time is 120 h.

fh scaling law only describes the inner part of the vortex ␤ ϭ max , e Ha wind pro®le. According to Fiorino and Elsberry (1989), 0 the strength and radius of maximum wind have little where we have used the e-folding distance of the to- in¯uence on the drift speed when a variety of vortex pography in the east±west direction for the terrain slope shapes are considered. It is the outer parts of the vortex calculation. The estimated drift speed in Fig. 5 is ap- that act with ␤ to create a steering current to advect the proximately 5.4, 5.4, 5.4, and 3.7 m sϪ1 in (a), (b), (c), inner vortex. Thus, the scaling law is valid for vortices and (d), respectively. The drift speeds in Fig. 5, however, with speci®c wind pro®les. are not in agreement with the Smith scaling law. This With the drift speed related to the ``local northwest- probably has to do with the portion of the topography ward'' propagation, it is natural to ask whether the vor- that is felt by the vortex in Fig. 5; this varies as we tex will reach the top of the island and remain locked change our rmax parameter. It appears that the application there. We have not observed such phenomena in our of the Smith scaling law to our calculation is quite sen- simulations. Carnevale et al. (1991) also did not ®nd sitive to the degree that the local ␤e effect is felt by the this behavior for vortex motion on a conical hill. They vortex. The Smith scaling law applied to our discussion pointed out that the vortex ¯ow over the peak would indicates that vd ϳ ͙␤evmax. The dynamics of the Smith cause the development of an anticyclonic circulation scaling law indicate that the drift speed vd is proportional over the peak. This would block the movement toward to the asymmetric vorticity ␨a. The ␨a results from the the peak and cause the vortex to move around the peak local vorticity advection (order of ␤evmax) and is de- in a clockwise direction, as they observed. From another stroyed by the drift of the vortex. Thus, we have point view the vortex does not climb to the top because the length scale of horizontal gradient of topography ␤␷e max ␷daϳ ␨ ϳ , and ␷ dϳ ͙␤␷ emax. approaches zero near the top. Thus, a vortex cannot ␷ d continue its drift toward the peak of the island.

Note that the parameter vmax that is used in the Smith We have also performed linear calculations for our

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FIG. 4. Similar to Fig. 3 except for the sensitivity of the track with respect to the hmax

parameter. The topographic parameters are a0 ϭ 400 km, and b0 ϭ 600 km. The vortex Ϫ1 parameters are rmax ϭ 100 km and vmax ϭ 20ms . The hmax is 1500, 2500, 3500, and 4500 m in (a), (b), (c), and (d), respectively.

f-plane experiments. Our linear calculations indicate a remain circular) were also studied by Carr and Williams slower drift speed circulating the topography (not shown (1989) with the bounded Rankine vortex with zero basic here). The slower speed in the linear case may be due state vorticity gradient. Smith and Montgomery (1995) to dispersion effects. It is the nonlinear effect that holds generalized Carr and Williams' 1989 solution to the case the vortex together and protects the vortex from strong of unbounded Rankine ¯ow. On the other hand, Drit- dispersion in the island-circulating path. The dynamical schel (1998) and Kuo et al. (1999) suggested that the basis for the dispersion-resistant properties in a baro- nonlinear dynamics hold all the vortex Rossby waves tropic vortex can be discussed in terms of the so-called moving in phase at the same phase speed and maintain ␤ Rossby number (McWilliams and Flierl 1979; Sutyrin the elliptical vortices from dispersion. The dynamics of and Flierl 1994). The ␤ Rossby number is the ratio of barotropic vortex Rossby waves that propagate both ra- the nonlinear terms in the vorticity equation to the linear dially and azimuthally on the radially continuous wave- vortex Rossby wave dispersion term. The ␤ Rossby guide of the vortex vorticity gradient was thoroughly number in our application can be de®ned as studied by Montgomery and Kallenbach (1997). ␷ R ϭ max . ␤ 2 ␤e rmax b. Sphere experiments When this dimensionless parameter drops below unity, The tracks for the experiments on the sphere with a

a stronger ␤e dispersion and energy radiation from the T192 resolution global shallow-water model are pre- vortex is expected. When the number is of order of unity, sented in Fig. 6. The topography contours in all the the nonlinear effect is important, and the vortex can global shallow-water model experiments start at 500 m

hold itself against the dispersion. With the rmax of the and increment by 500 m. The vortex parameters are rmax Ϫ1 order 100 km in our speci®ed vortex, it requires a vmax ϭ 100 km and vmax ϭ 20ms . The topography is of a few meters per second for the nonlinear effect to centered at 45Њ north latitude, and (a) a 0 ϭ 100 km, b 0 be important. The nonlinear dynamics are thus impor- ϭ 150 km, hmax ϭ 4500 m; (b) a 0 ϭ 50 km, b 0 ϭ 150 tant for tropical cyclone strength vortices. The nonlinear km, hmax ϭ 4500 m; (c) a 0 ϭ 100 km, b 0 ϭ 150 km, dynamics of ``asymmetry damping'' (to make the vortex hmax ϭ 2500 m; and (d) a 0 ϭ 50 km, b 0 ϭ 150 km, hmax

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FIG. 5. Similar to Fig. 3 except for the sensitivity of the track with respect to the vortex

scale rmax. The topographic parameters are a0 ϭ 200 km, b0 ϭ 300 km, and hmax ϭ 3500 m. Ϫ1 The vortex parameter is vmax ϭ 20 m s . The rmax is 100, 200, 300, and 400 km in (a), (b), (c), and (d), respectively.

ϭ 2500 m. The black dot represents the center position the T96 resolution. Note that the track dynamics due to

(potential vorticity maximum) of the vortex. The time the variation of b 0 are not equivalent to the track dy- interval between adjacent black dots is 24 h. The vortex namics due to the variation of a 0 on the sphere. Figure initial position is indicated by the ϫ symbol. The total 7 shows the sensitivity of the track to the topographic integration time is 216 h. Comparing Fig. 6a with the parameter b 0. The topography is at 25Њ latitude and hmax f plane results, we see that the island-circulating path ϭ 2500 m. The topographic e-folding distances are (a) may also occur in the presence of a planetary vorticity a 0 ϭ 600 km, b 0 ϭ 600 km, (b) a 0 ϭ 600 km, b 0 ϭ gradient. Although the topographic slope is larger in the 500 km, (c) a 0 ϭ 600 km, b 0 ϭ 400 km, and (d) a 0 ϭ case of Fig. 6b, the vortex still breaks away from the 600 km, b 0 ϭ 300 km. Figure 7 indicates that the vortex island after a partial clockwise island-circulating track. trapping is not as sensitive to the parameter b 0 as to the The vortices in Figs. 6c and 6d break away from the parameter a 0. Similar to Fig. 7, Fig. 8 shows the sen- island after some clockwise in¯uence. It is clear from sitivity of the track to the latitude of the island. The

Fig. 6 that a higher mountain favors the existence of an topographic parameters are a 0 ϭ 300 km, b 0 ϭ 600 km, island-circulating track. The presence of the global ␤ and hmax ϭ 2500 m. The vortex parameters are rmax ϭ Ϫ1 effect, contrary to the local ␤e effect, tends to break the 300 km and vmax ϭ 20 m s . The latitude of the island vortex away from the topography. is 15Њ,25Њ,35Њ, and 45Њ in (a), (b), (c), and (d), re- To understand the dynamics of vortex trapping on the spectively. Figure 8 shows that a higher latitude island sphere, we have performed extensive T96 experiments favors the clockwise island-circulating track. Moreover, with larger topographic parameters a 0, b 0, and a larger the drift speed in the island-circulating track is slower rmax. The larger parameter range is chosen to allow ex- in the low latitude case. This may be due in part to the tensive experiments with less cost and to avoid reso- fact that the Coriolis force is larger at higher latitudes lution problems. The scale parameters are larger than in addition to the slower ␤-induced speed in high lati- would be expected in real tropical cyclones, but a com- tudes. Due to the complication of the ␤ effect, the drift parison of the T96 and T192 solutions shows that the speed does not appear to be in agreement with the Smith fundamental dynamics are not altered by using larger scaling law. scales. The following experiments are performed with Figure 9 shows the sensitivity of the track to the

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FIG. 6. The track for the experiments on the sphere with global shallow-water model. The vortex parameters are rmax ϭ 100 km and vmax Ϫ1 ϭ 20ms . The topography is centered at 45Њ north latitude, and (a) a0 ϭ 100 km, b0 ϭ 150 km, hmax ϭ 4500 m; (b) a0 ϭ 50 km, b0 ϭ

150 km, hmax ϭ 4500 m; (c) a0 ϭ 100 km, b0 ϭ 150 km, hmax ϭ 2500 m; and (d) a0 ϭ 50 km, b0 ϭ 150 km, hmax ϭ 2500 m. The topography contours start at 500 m and increment by 500 m. The black dot represents the center position (potential vorticity maximum) of the vortex. The time interval between adjacent black dots is 24 h. The vortex initial position is indicated by the ``ϫ'' symbol. The total integration time is 216 h.

vortex intensity vmax for the island centered at 25Њ lat- about the same on the vortex motion. Thus, the trapping itude. The topographic parameters are a 0 ϭ 300 km, b 0 of the vortices is not sensitive to the vmax of the hurri- ϭ 600 km, and hmax ϭ 2500 m. The vortex scale pa- cane-strength vortex. rameter is rmax ϭ 300 km. The vmax is 10, 20, 30, and The sensitivity of the track with respect to hmax for 40msϪ1 in (a), (b), (c), and (d), respectively. As dis- the island centered at 25Њ latitude is shown in Fig. 10. cussed in Chan and Williams (1987) and Smith et al. The topographic parameters are a 0 ϭ 300 km and b 0 ϭ (1997), the ␤-induced vortex motion speed (which tends 600 km. The vortex parameters are rmax ϭ 300 km and Ϫ1 to break the vortex from the topography) increases with vmax ϭ 20ms . The value of hmax is 1000, 2000, 3000, increasing vmax. It appears that the effect of the mountain and 4500 m in (a), (b), (c), and (d), respectively. Figure also increases with vmax so that the net effect would be 10 suggests that higher mountains are more favorable

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FIG. 7. Similar to Fig. 6 except for the sensitivity of the track with respect to the topographic parameter b0. The topography

is at 25Њ north latitude and hmax ϭ 2500 m. The topographic e-folding distance is (a) a0 ϭ 600 km, b0 ϭ 600 km; (b) a0 ϭ 600

km, b0 ϭ 500 km; (c) a0 ϭ 600 km, b0 ϭ 400 km; and (d) a0 ϭ 600 km, b0 ϭ 300 km, respectively. for island-circulating tracks. Figure 11 shows the sen- southeast of the island. Figure 12 is the same as Figs. sitivity of the track to the vortex scale rmax for the island 9b, 9d, and 11d except that the initial position of the centered at 25Њ latitude. The topographic parameters are vortex is displaced 3Њ to the north. By comparing Figs. a 0 ϭ 300 km, b 0 ϭ 600 km, and hmax ϭ 2500 m. The 12a and 12b with Figs. 9b and 9d, we ®nd that the Ϫ1 vortex parameter is vmax ϭ 20ms . The value of rmax general properties of vortex trapping for small vortices is 300, 400, 500, and 600 km in (a), (b), (c), and (d), (rmax ϭ 300 km) are similar except for some details in respectively. We observed from Fig. 11 that the vortices position. By comparing Fig. 12c with Fig. 11d, we ®nd with smaller radii of maximum wind are more likely to that the track of a larger vortex (rmax ϭ 600) is quite follow the island-circulating track. For vortices with a chaotic in the sense that a slight change in the initial comparable ratio of rmax to a 0, we see trapping in the f position (i.e., close to a 3⌬x difference in our experi- plane runs but no effect on the sphere. ments) will result in a drastically different track. Figure We have so far carried out our spherical experiments 12 indicates that the sensitivity of the impinging position with the impinging position of the vortex always to the of the vortex on the topography on the sphere experi-

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FIG. 8. Similar to Fig. 6 except for the sensitivity of the track with respect to the latitude of the island. The topographic

parameters are a0 ϭ 300 km, b0 ϭ 600 km, and hmax ϭ 2500 m. The vortex parameters are rmax ϭ 300 km and vmax ϭ 20 m sϪ1. The latitude of the island is 15Њ,25Њ,35Њ, and 45Њ north in (a), (b), (c), and (d), respectively. ments. This is quite different from our f-plane exper- the ratio of the two ␤-drift effects. The circulating and iments, which show that the trapping is not sensitive to noncirculating cases are separated by examining the the impinging position. This sensitivity and the bifur- track and the end position of the vortex. Figure 13 sug- cation point for the trapped versus untrapped vortices gests that the circulating mode is favored when a 0/rmax should be studied with a much higher resolution model. Ն 1.7␤/␤e ϩ 0.5. In general, the trapping and clockwise We have summarized all the T96 experiments in a circulating path in the presence of a planetary vorticity two-dimensional parameter space. The abscissa in Fig. gradient will occur if the scale of the topography is

13 is the nondimensional parameter ␤/␤e, and the or- greater than the vortex radius of maximum wind and if dinate is a 0/rmax, the ratio of the horizontal width of the the planetary ␤ parameter is less than the topographic topography in east±west direction to the vortex scale in ␤ parameter. The ␤e effect is to trap the vortex in the terms of rmax. The parameters are chosen on the ground vicinity of topography and to produce an island-circu- that the island-circulating track is most likely due to the lating mode. When ␤ is smaller than ␤e, the vortex will topographic ␤-drift effect. The parameter ␤/␤e measures be trapped. On the other hand, the vortex scale has to

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FIG. 9. Similar to Fig. 6 except for the sensitivity of the track with respect to the vortex intensity vmax for the island centered

at 25Њ north latitude. The topographic parameters are a0 ϭ 300 km, b0 ϭ 600 km, and hmax ϭ 2500 m. The vortex scale parameter Ϫ1 is rmax ϭ 300 km. The vmax is 10, 20, 30, and 40 m s in (a), (b), (c), and (d), respectively. be comparable to (if not smaller than) the scale of the in our simulations is very different from the linear grav- topography for the vortex to feel the existence of the ity wave speed ͙gH. Our phenomena involve the prop- topography. The vortex trapping is not sensitive to vmax agation of a ®nite amplitude potential vorticity anomaly, which is why the coordinates in Fig. 13 do not depend while linear Kelvin waves have zero potential vorticity. on vmax. The nondispersive, trapping mode is also clearly seen in rigid lid single-layer quasigeostrophic calculations, which do not support Kelvin waves (Carnevale et al. 4. Concluding remarks 1988). We have also performed calculations with a dif- The nondispersive, clockwise, island-circulating ferent structure parameter b ϭ 0.25 in (2.7) and with a mode of hurricane-like vortices is qualitatively similar shallower equivalent depth. A rich variety of tracks were to the coastally trapped waves (e.g., Gill 1977) and simulated. The clockwise, island-circulating modes in Kelvin waves that also possess the nondispersive clock- these simulations were qualitatively similar. With a shal- wise island-circulating feature. However, the drift speed lower mean depth, we observed a slower circulating

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FIG. 10. Similar to Fig. 6 except for the sensitivity of the track with respect to the hmax parameter for the island centered at

25Њ latitude. The topographic parameters are a0 ϭ 300 km and b0 ϭ 600 km. The vortex parameters are rmax ϭ 300 km and Ϫ1 vmax ϭ 20 m s . The hmax is 1000, 2000, 3000, and 4500 m in (a), (b), (c), and (d), respectively. motion, which is consistent with the divergent topo- observations by Doppler on Wheel (Gall et al. 1998; graphic ␤-drift effects. A thorough analysis of the di- Wurman and Winslow 1998) revealed the existence of vergence effect on the shallow-water model results can intense, small-scale spiral bands for landfall hurricanes. be found in Zehnder (1993). These spiral bands have spatial scales on the order of We have observed wake and secondary low formation subkilometer to 10 km and extend through depths of 1 in some of our calculations with a mountain peak height to 5±6 km. It appears that the mesoscale features (hor- of 4500 m (i.e., M ϭ 0.9). These lows sometimes also izontal length scale smaller than the terrain scale) as- followed the island-circulating track and at times sociated with typhoon landfall are very complicated, and merged with the main vortex. The trapping and clock- these multiple scale structures are not well resolved by wise circulating path does not seem to be affected by observations. No attempt has been made to explore the these mesoscale features. A detailed comparison with details of mesoscale features of secondary lows in this Smith and Smith (1995) cannot be made due to the paper. Rather, the focus has been on the major features inadequate resolution used in our calculations. Recent in the vortex track in response to the terrain. There are

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FIG. 11. Similar to Fig. 6 except for the sensitivity of the track with respect to the vortex scale rmax for the island centered at 25Њ latitude. Ϫ1 The topographic parameters are a0 ϭ 300 km, b0 ϭ 600 km, and hmax ϭ 2500 m. The vortex parameter is vmax ϭ 20 m s . The rmax is 300, 400, 500, and 600 km in (a), (b), (c), and (d), respectively. problems, such as the development mechanism of the island-circulating mode is consistent with the simula- secondary low, severe downslope winds, boundary layer tions of Zehnder. Since the spatial scale of the Sierra structure, and frictional drag effects in landfalling hur- Madre is much greater than the vortex scale, the de- ricanes, that deserve further study. ¯ections produced by the island-circulating mode can Zehnder (1993) studied the in¯uence of large-scale be quite signi®cant. In fact, Zehnder argued that the topography, such as the Sierra Madre, on barotropic mechanism might be important in forecasting hurricane vortex motion with an equatorial ␤-plane channel mod- motion in the eastern north Paci®c and the western Gulf el. He found that the motion of vortices is de¯ected by of Mexico. A study on the propagation of wintertime the large-scale topography. As clearly pictured in his synoptic scale low-level circulation features in the vi- Figs. 5 and 8, a vortex that approaches the topography cinity of a mountain range in terms of topographic Ross- from the east is accelerated and de¯ected toward the by waves can be found in Hsu (1987). Hsu argued that south, and a vortex initially west of the mountain re- the steering effect of topography upon atmospheric cir- curves and moves toward the east. It is clear that our culations is restricted to the lower atmosphere due to

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FIG. 12. Same as (a) Fig. 9b, (b) Fig. 9d, and (c) Fig. 11d except that the impinging position of vortex to the island is 3Њ farther north.

the strong vertical strati®cation in wintertime. However, not present. Our results indicate that vortices with small- the dominant diabatic processes in tropical cyclones can er radius rmax and islands with a higher peak height, with greatly increase the vertical coupling. The topography- a wider east±west scale and which are at a higher latitude induced steering may not be restricted to the lower at- favor vortex trapping into a clockwise, island-circulat- mosphere for the hurricane. ing track. To be consistent with the results, we need to

Although the calculations contain no mean ¯ow, our consider both the scale parameter rmax/a 0 and the time- discussion can shed some light on the situation when scale of vortex crossing the topography by advection. mean ¯ow is present. Let us consider a situation where The scale of the vortex must be smaller than the scale the mean steering ¯ow does not vary appreciably over of the island for the vortex to be trapped. Similarly, the the spatial scale of the island. Without the spatial var- time it takes the vortex to drift directly across the island iation of the mean ¯ow, the ``environmental ␤-drift'' topography must be longer than the timescale of the due to the potential vorticity gradient of mean ¯ow is earth's rotation 1/ f. This argument can be written in

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raphy as a ®xed positive potential vorticity anomaly, which comes only from the mass ®eld. Our vortex-trap- ping problem can then be interpreted as an interaction of a time-varying potential vorticity anomaly with a permanent potential vorticity anomaly. Our results sug- gest that small, strong vortices become trapped and cir- culate around the ``permanent vortex'' (or topographic feature). Although the trapped, circulating mode does

not depend on vmax, the vortex has to be strong for the nonlinear effects to maintain the vortex against disper- sion. This maintenance of a small, strong vortex em- bedded and circulating within a larger vortex may be of importance on quite different spatial scales in prob- lems such as the eyewall mesovortices embedded in the cyclonic circulation of tropical cyclones (Stewart and Lyons 1996; Stewart et al. 1997; Lander et al. 1999).

FIG. 13. Summary of numerical experiments with nondimensional In such a problem, trapping and weak dispersion re- parameters. The abscissa is the nondimensional parameter ␤/␤e, quires a large ``vortex ␤ Rossby number'' with the radial where ␤e is the equivalent ␤ effect of the topography. The ordinate vorticity gradient of the circular basic state vortex re- is a0/rmax the ratio of topographic horizontal width to the vortex scale. placing ␤, the eddy velocity replacing vmax, and the eddy The island-circulating experiments are indicated by ᭡ (vmax ϭ 40 m Ϫ1 Ϫ1 length scale replacing rmax (Enagonio and Montgomery s ) and ࡗ (vmax ϭ 20ms ) symbols. The non-circulating tracks Ϫ1 Ϫ1 2001). The mechanism may also be relevant to meso- are indicated by ϫ (vmax ϭ 40ms ) and ϩ (vmax ϭ 20ms ) symbols. scale systems in¯uenced by topography. The results have oceanographic applications as well. Oceanogra- terms of a Rossby number. The topographic de¯ection phers have studied similar dynamics for the effects of is important only when both the appropriate Rossby underwater meridional ridges on vortices (e.g., Kamen- number and the scale parameter rmax/a 0 are smaller than kovich et al. 1996). Vortices must be strong enough to unity. The argument is in agreement with the obser- maintain themselves against dispersion and also be in vations that the more intense and rapidly moving hur- a region with weak currents. The coherency of the vor- ricanes moved directly cross over the Taiwan barrier. tices in the ocean allows thermal and salinity anomalies With simple dynamical model calculations, our intent to be carried far downstream. is not to underestimate the importance of the moist phys- ics, large-scale steering, and baroclinic effects but rather Acknowledgments. This research was supported by to isolate the fundamental barotropic aspects of the im- grant NSC88-2111-M002-017 AP1 and NSC88-2625- pact of topography on vortex movement. This study Z002-029 from National Research Council of Taiwan, compliments previous work on propagation of baro- NSF grant ATM-9525755, and P.E. 060435N from the tropic vortices over topography (e.g., Carnevale et al. Of®ce of Naval Research through the Naval Research 1991), and we have extended the track dynamics to Laboratory. We would also like to thank Drs. M. T. include both local topography ␤e and global ␤ effects Montgomery and G. Carnevale and an anonymous re- for hurricane-like vortices. When a vortex approaches viewer for useful comments in the paper. a low-latitude island like Taiwan, the vortex translation speed and the scale of Taiwan may not be right for the occurrence of a complete island-circulating track. On REFERENCES the other hand, our simulations suggest the possibility Adem, J., 1956: A series solution for the barotropic vorticity equation of a vortex breaking away from the island after follow- and its application in the study of atmospheric vortices. Tellus, ing a partial clockwise island-circulating track for slow- 8, 364±372. Bender, M. A., R. E. Tuleya, and Y. Kurihara, 1987: A numerical moving small hurricane-like vortices. This partial clock- study of the effect of island terrain on tropical cyclones. Mon. wise island-circulating track can also be seen from time Wea. Rev., 115, 130±155. to time in the typhoon tracks cross Taiwan. Examples , I. Ginis, and Y. Kurihara, 1993: Numerical simulations of the of such tracks are shown in Fig. 1 for Typhoon Sarah tropical cyclone±ocean interaction with a high-resolution cou- (1989) and Typhoon Bess (1952). The general agree- pled model. J. Geophys. Res., 98, 23 245±23 263. Brand, S., and J. Blelloch, 1974: Changes in the characteristics of ment of the nondispersive local ␤e vortex dynamics with typhoons crossing the island of Taiwan. Mon. Wea. Rev., 102, the observations, however, may also be coincidental. 130±155. With the very limited upper-air observations during ty- Carnevale, G. F., G. K. Vallis, R. Purini, and M. Briscolini, 1988: phoon landfall, it is dif®cult to distinguish the ␤ effect Propagation of barotropic modons over topography. Geophys. e Astrophys. Fluid Dyn., 41, 45±101. from the environmental steering effect. , R. C. Kloosterziel, and G. J. F. van Heijst, 1991: Propagation Our results can also be interpreted in terms of po- of barotropic vortices over topography in a rotating tank. J. Fluid tential vorticity conservation. We can view the topog- Mech., 233, 119±139.

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