December 1969 K. Tsuchiya 457

The Clouds with the Shape of Karman Vortex Street

in the Wake of Cheju Island, Korea

By Kiyoshi Tsuchiya

Japan Meteorological Agency

(Manuscript received 9 May 1969, in revised form 14 October 1969)

Abstract

The properties of the clouds with the shape of Karman vortex street in the wake of Cheju Island, Korea are studied using satellite pictures which are taken approximately four hours apart. The results indicate that non-dimensional numbers and physical quantities obtained from the cloud vortex street satisfy theoretically as well as experimentally obtained criteria of Karman vortex street. The equation of the displacement of the cloud vortex street is derived through drag consideration. The displacement speed obtained from this equation agrees fairly well with the observed value. The speed is approximately 76 % of the undisturbed flow when the ratio between the spacing of two rows and that of individual vortex is 0.332.

It is foundd that such a cloud vortex street appears in the wake of Cheju Island from late autumn to early spring when steady northerly wind of more than moderate intensity (above 10 kts) prevails and a well defined inversion is found at around 1000 meters or at about half the height of the mountain located in the center of the Island.

1. Introduction

The existence of mesoscale atmospheric eddies in the wake of an island has been revealed by satellite pictures. Based on TIROS pictures Hu-bert and Kreuger (1962) first made an analysis on these mesoscale atmospheric eddies and suggested three possible formation mechanisms, i. e. purely mechanical, inertial oscillation and inertial insta-bility. Further study was made by Chopra and Hubert (1965) who suggested that a particular type of mesoscale eddies manifested in the form of clouds in satellite pictures could be explained by the theory of Karman vortex street. They made an estimation of the displacement speed and various parameters based on the ratio between the distance of vortex rows and that of successive vortices, but no verification was possible for them due to lack of data showing later configuration of cloud vortex.

Through the study of daily APT pictures as well as AVCS pictures in the Far East, it is found that the clouds with the shape of Karman vortex street appear once in a while in the wake of Cheju Island, Korea from autumn to spring. Taking advantage of the pictures taken by two meteorological satellites ESSA 7 and 8 approx-imately 4 hours apart on March 5, 1969, the

properties of this cloud vortex street is studied in reference with the theoretical and experimental

Fig. 1. ESSA 8 APT picture showing clouds with the shape of Karman vortex street in the

wake of Cheju Island, Korea. 41 h 24 m 25s GMT, March 5, 1969.

458 journal of the Meteorological Society of Japan Vol. 47, No. 6

results of Karman vortex street.

2. Satellite pictures showing the cloud pattern with the shape of Karman vortex street and

synoptic situation

Fig. 1 is an APT picture taken at 01 h 24 m 25 s GMT, March 5, 1969 showing a cloud

pattern similar to Karman vortex street in the wake of Cheju Island, Korea. Vortex streets can be well recognized to the distance of nearly 600 km in the wake of the island. Later configuration of the vortex streets is shown in Fig. 2 (a) and (b), ESSA 7 AVCS pictures taken at 05 h 06 m 52s and 05 h 11 m 13 s GMT respectively on the same day. Original gridding of the pictures of Fig. 2 made at NESC, which is indicated in white dot-ted lines, was off approximately 1 degree. Since the land marks such as Kyushu, Cheju Island and Korean Peninsula were clearly visible in the

pictures, there was no difficulty in regridding them correctly. Comparing Figs. 1 and 2, we can easily see that the vortices have moved on the average approximately 0.85 degrees in great circle dis-tance during the time of these pictures. It is

also seen that the young vortex indicated by "A"

in Fig. 1 has fully developed in Fig. 2 a and b.

Fig. 3. The map of Cheju Island indicating the location of radiosonde and surface ob- serving stations.

Fig. 2. ESSA 7 AVCS pictures showing later configuration of the clouds shown in Fig. 1. The

picture taking times are: (a) 05 h 06 m 52 s and (b) 05 h 11 m 13 s GMT March 5, 1969.

Deccmber 1969 K. Tsuchiya 459

Topographical feature and the location of ob-

serving stations in the island are indicated in

Figs. 3 and 8. It is considered that the result of

laboratory experiments using simple shaped ob-

stacles may be applicable to this case due to a

rather simple shape of the island.

Fig. 4 is the vertical structure of the atmos-

phere above Mosulpo which is located in the

western edge of the island and two other Ja-

panese stations, Kadena and Naze, located down

stream. The sounding curve of Mosulpo shows

three layers of temperature inversion with the

base of the lowest one at approximately half the

height of the mountain in the center of the

island. The sounding curves of Kadena and Naze

also indicate a well defined inversion layer at

the level of 850 mb. A similar feature is also

found in the analyses of other cases, and it is

considered that the existence of an inversion at

about half the height of the mountain is one of

the necessary conditions for the formation of

such cloud vortex street. The vertical distribution

of dew point suggests that the thickness of vor-

tex cloud is approximately 200 meters.

Fig. 4. Radio sonde observations at Mosulpo, Korea (47187). Kadena (47931) and Naze (47909), Japan. Solid and broken lines are air temperature and dew point respectively.

(a) 850 mb chart at 00 GMT, March 5, 1969. (b) The surface weather map at 00 GMT, March 5, 1969.

Fig. 5.

460 Journal of the Meteorological Society of Japan Vol. 47, No. 6

The surface and 850 mb charts of 00 GMT, March 5, 1969, an hour before the picture time of Fig. 1 are indicated in Fig. 5 (a) and (b). The

general current over the area where the vortex street is seen is NNW or N which is nearly

perpendicular to the major axis of the island and parallel to the orientation of the vortex street. Farther down stream, the general flow curves anticyclonically near Okinawa Islands. This change in the direction of the general flow affects the orientation of the vortex street there. The surface chart of six hours later which approximately cor-responds to the picture time of Fig. 2 (a) and (b) is indicated in Fig. 6. It can be seen that winds

developed anticyclone to the west of Korea and a cyclone to the east of Japan with northerly wind of more than moderate intensity over the East China Sea.

3. The parameters of vortex cloud street.

3.1 The ratio of the spacing of the vortex rows and that of successive vortices

Defining a and b as indicated in Fig. 7, K~rman (1912) theoretically proved that a vortex street can be stable only when b/a is 0.281 in case of no side boundary fluid. However further laboratory experiments show that vortex street can be formed when the ratio is within the range of 0.28 and 0.52 (Chopra and Hubert).

Fig. 7. Schematic representation of the parameters

of Karman vortex street.

Fig. 6. The surface weather map at 06 GMT,

March 5, 1969.

have become weaker. The wind aloft observations made at Japanese aerological stations at 06 GMT indicate a similar tendency below 700 mb level and 1200 GMT wind aloft observations indicate further weakening winds, thus the average speed of the general current on which the vortex street is imbedded is estimated to be 18 kts.

The surface observation indicates that the cloud forming vortex street is mostly strato-cumulus which coincides with the analysis of radiosonde data. In reference with other cases, it is found that the pressure patterns shown in Figs. 5 and 6 are typical ones when a vortex cloud street is formed in the wake of Cheju Island, i. e. a well

The ratio of b to a for this case is indicated In Table 1. The number in the left column is taken in such a way that it increases southward. The difference in the values obtained from Figs. 1 and 2 is small although the sizes of the vortices and lengths of b and a in Fig. 2 are slightly larger than those of Fig. 1. The average value of b/a obtained from Fig. 1 and 2 is 0.332.

3.2 Vortex shedding period, life time and displacement speed of the vortex streets

The average displacement speed C of the vortex streets during the period of picture times of Figs. 1 and 2 is found to be 7.0 m/s, which is about 76 % of the average speed of the general current. With the knowledge of C, the shedding period r is easily computed from Eq. (1).

Substituting the average distance of a shown in Table 1, we obtain the shedding period v as 4.4 hours, which means the life time of the cloud vortices shown in Figs. 1 and 2 is considerably long.

It is possible to derive an equation which gives C as a function of b/a. Defining U, k, D and

p to be the speed of the undisturbed flow, circu-

December 1969 K. Tsuchiya 461

Table 1. Spacing of vortex rows (a) and that of individual vortices (b) and the ratio between them.

lation of the vortex, drag due to a vortex wake and the density of the fluid, respectively, the following equations hold for Karman vortex street

(Milne-Thomson, 1960).

For the special case of A=B, or b/a =0. 381, Eq. (7) becomes,

Now Eq. (7) gives real roots when b/a satisfies the following criterion.

A dimensionless parameter, drag coefficient CD,

is defined by,

where d is a cross stream diameter of an ob-stacle. Then from Eqs. (3) and (4), the following relation is obtained.

Substitution of previously obtained numerical

values of the parameters yields,

Eqs. (5) and (6) lead to the following equation.

Laboratory experiments indicate that of two real roots only the larger one applies. This is understandable from a physical requirement that the displacement speed should increase according to the decrease of b/a finally reaching that of the undisturbed flow speed itself in the limit of b/a = 0. Substitution of b/a = 0 into Eq.

(7) yields 0 and 1, the latter value means the displacement speed is equal to that of the undis-turbed flow. Theoretical ratio, b/a=0.281 substituted into Eq.

(7), gives 0.81 for the larger root while the labo-

ratory observed value by Birkoff and Zarantonello

(1957) was about 0.85. From these facts it is considered that Eq. (7) can be used to estimate the displacement speed of cloud vortex streets for different ratios of b/a less than 0.427.

3.3 Circulation, vorticity and tangential velocity

For the previously obtained numerical values of U, C and b/a, Eq. (2) gives 512 x 103 m2 sec-1

462 Journal of the Meteorological Society of Japan Vol. 47, No. 6

as the value of the circulation. If an assumption is made that the size of the vortex is approxi-mated by the cloud vortices, then the tangential velocity can be obtained. From the average dia-meter of the first two vortices in Figs. 1 and 2,

(51 km), the tangential velocity and vorticity are obtained to be 3.2 m/sec or 34.6% of the undis-turbed flow speed and 2.5 x 10 -4sec-1 respectively, the latter corresponds to a typical value of that of mesoscale atmospheric eddies.

4. Drag coefficient (CD)

Several investigators have estimated drag coe-fficient. Table 2 {after Shlichting (1962)} indi-cates the values of CD obtained through labo-ratory experiments for the range that the for-mation of Karman vortex streettakes place. On the other hand, through a numerical experiment, Hirota and Miyakoda (1965) obtained 1.05 when Karman vortex street was formed at Reynolds Number Re =100 behind a circular cylinder. Their value agrees well with the one obtained by Payne (1958), Roshko (1954) and also the one obtained from Eq. (10) derived by Imai

(1957).

The values obtained by the latter investigators

are a little smaller than those shown in Table 2. It is shown in the previous section that CD can

be estimated from Eq. (6). The representative cross stream dimension of the island d may be approximated by Eq. (12).

Generally speaking h1 and h2 should be the sea surface and the base of the inversion layer, but

there are certain questions in this case. Fig. 4 indicates a thin isothermal layer above the lowest

layer characterized with dry adiabatic lapse rate

while in the upper part there are three inversion

Fig. 8. Cross section of Cheju Island which is per-

pendicular to NNw wind. Stippled narrow strip is the layer with relative humidity

above 70% at 00 GMT, March 5, 1969.

Table 2. Drag coefficient (CD) of different obstacles for Reynolds number (Re) in the range of which Karman vortex streets are formed (after Schlichting

1962).

Table 3. CD obtained from b/d for various characteristic dimension d defined by Eq. (12).

December 1969 K. Tsuchiya 463

layers. The vertical wind distribution observed at Mosulpo located at the south-western edge of the island suggests the existence of an eddy motion between approximately 900 and 1300 meters as is shown in Fig. 8. On the other hand, the es-timated thickness of the cloud is thin, at most 350 meters. In spite of these facts, if an assumption is made that d is the average along the major axis between the sea surface and the level of the highest inversion (1510 m), d is 28.3 km which gives 1.3 as the value of CD. Since d depends on the integration range while CD is determined from d, the values for different integration ranges are shown in Table 3 for com-

parison. In reference with the values obtained by many investigators through laboratory experi-ments and considering the fact that the value of CD for an obstacle of an irregular shape would be larger than a simple shaped one like a cylin-der or a sphere, the probable value would be somewhere between 0.9 and 1.5.

5. Reynolds Number (Re), Strouhal Number (S), kinematic eddy viscosity (v) and Lin's Number

( ) Based on detailed experiments, Blenk et al (19

35) pointed out that Karman vortex street is formed when Reynolds Number defined by Eq. (13) is in the following range: 60 <Re <5000.

The configuration of vortex street, however, is

widely defferent from each other according to the

values of Reynolds Number as is shown in Fig.

9 which is due to Homann (1936) or in other textbooks like Bachelor's (1967). Since the value of v is not known in this case, an attempt is made to estimate Re indirectly. According to Roshko (1954), Strouhal Number S defined by Eq. (14) uniquely depends on Re.

Substituting the previously obtained values of d (hereafter d is used for d), U and z, we obtain 0.194 as the value of S. The corresponding Re and CD obtained from Roshko's curve are 250 and 1.01 respectively. It is interesting to see a great resemblance between the shape of vortex streets of Re' s 225 and 281 in Fig. 9 and those in the satellite pictures of Fig. 1 and 2. The value of v obatined from Eq. (13) for Re =250 is 1.05 x 107 cm2/sec which seems slightly smaller than that obtained from Lin's criterion (Lin, 1959). Lin found that Karman vortex street is formed behind a cylindrical obstacle when nondimensinal number defined by Eq. (15) is in the range: 10-3< <2.5 10-3.

Substitution of previously obtained values of v, U and gives 0.78 x 10-3 as the value of which is slightly smaller than Lin's criterion. On the other hand his criterion requires to be : 1.09 x 107< v < 2.73 x 107 cm2/sec from which the range of Re becomes 95 < Re < 241 for U and d of this case. Summarizing the analyses made so far, the probable values of the ranges

Fig. 9. Karman vortex street under different Reynolds Numbers (After Homann, 1936)

464 Journal of the Meteorological Society of Japan Vol. 47. No. 6

of nondimensional numbers Re, S, and kine-matic eddy viscosity will be as follows :

It is worthwhile to notice that the above-men-

tioned numbers are in the ranges found in the

laboratory experiments for Karman vortex street.

6. Concluding remarks

If the previous analyses are acceptable, they

lead to the following conclusions. The clouds

shown in Figs. 1 and 2 are atmospheric analogue

of Karman vortex street. Although there is a

little discrepancy in the values of nondimensional

numbers of cloud vortex street compared with

those obtained through laboratory experiments

for two dimensional flow , an interesting fact is that they satisfy the criteria presented previously ,

thus the theory and the laboratory experimental

results are applicable in explaining the similar

atmospheric phenomena the scale of which is

several orders of magnitude larger . The life time of

the cloud vortex street is fairly long, attaining

more than a day when the situation is favorable . This fact suggests a practical and more efffective

application of satellite pictures for weather anal-

ysis and forecasting. The values of the parame-

ters obtained in this analysis may be applicable

in dynamic equations.

Acknowledgements

The author is greatly indebted to Messrs . Shozo

Ito of Japan Meteorolgical Agency and Lester

Hubert, National Environmental Satellite Center, ESSA, U. S. A. for their valuable suggestions and

comments. His deep appreciation is due to Mr.

Vincent Oliver of NESC, ESSA, U. S. A, for the

kind arrangements to obtain ESSA 7 AVCS

pictures.

References

Bachelor, G.K., 1967: An introduction to fluid dynamics, Cambridge Univ. Press, 615 pp.

Birkoff, G. and E. H. Zarantonello, 1957: Wakes and cavities. New York Academic Press, 353 pp.

Blenk, H., D. Fuchs and L. Liebers, 1953: Uber die Messung von Wirbelfrequenzen. Luftfahrtforschung.

XII, 38-41. Chopra, K. P. and L. F. Hubert, 1965: Mesoscale

eddies in the wake of islands. J. of Atm. Sci. 22, 652-657.

Hirota, I, and K. Miyakoda, 1965: Numerical solution of Karman vortex street behind a circular cylinder. J. Meteor. Soc. Japan, 43, 30-41.

Homann, F., 1936: EinfluL3 grober Zahigkeit bei Stro- mung um Zylinder. Forsch. Ing. Wes., 7, 1-10.

Hubert, L, and A. F. Krueger, 1962: Satellite pictures of mesoscale eddies. Mon. Wea. Rev., 90, 457-463.

Imai, 1., 1957: Theory of bluff bodies. Univ. of Mar-

yland, Inst. Fluid Dyn. & App!. Math. Tech. Note. BN 104.

Karman TH. von and H. Rubach, 1912: Uber den Mechanismus des Fliissigkeits and Luftwiderstan-

des. Phys. Z. 13, 49-59. Lin, C. C., 1959: On periodically oscillating wakes in

the Oseen approximation. Studies in fluid mechanics. R. von Mises N. Y. Academic Press, 170-176.

Milne-Thomson, L.M., 1960: Theoretical hydrodynamics. 4th ed. Mc-Millan Co. N.Y., 660 pp.

Payne, R., 1958: Calculations of viscous flow past a circular cylinder. J. Fluid Mech., 4, 81-88.

Roshko, A.,1954: On the development of turbulent wakes

from vortex streets. NACA Rep. Schlichting, H., 1962: Boundary layer theory, (Translated

by Kestin J. 4 th ed.), Mc-Grow-Hill Co., 647 pp.

済州島の風下のKarmanう ず列状雲

土 屋 清

気象庁予報部,気象研究所(併 任)

約4時 間 の時間 間隔で撮影 した気象衛星エ ッサ7号 と8号 の写真に現われ た済 州島風下 のKarman鋤 うず列状 の雲

を解析 した結果,次 の ことがわか った.

雲 の うず列か ら得 られ た無次元パ ラメー ターは,こ れ まで2次 元流 の流体実験か ら得 られ たKarman鋤 うず列発 生

の限界値 を満足 してお り,Karmanう ず列 の理論で説明で きる.島 の抵抗 とうず列 のwakeに よる抵抗 を考慮 して,

うず列の間隔 と個 々の うずの間隔の比だけか ら,う ず列の移動速度を求め る計算式を導 いた.結 果は実況 とか な りよ

December 1969K.Tsuchiya 465

く合 い,比 が0.332の とき.一 般流の約76%で あ った.

済州島 の風下に,こ のよ うな雲 の うず列 のできるのは,晩 秋 か ら初春にかけ て,比 較的強 い(10ノ ッ ト以上)北 風

が持続 し,1000m付 近(島 の中央部 の山の高 さの半分)に 顕著な逆転のあ る時で ある.