˜_•¶ Memoirs

On the Negative Vorticity in a *

S. Syono, Y. Ogura, K. Gambo and A. Kasahara

(GeophysicalInstitute, University)

- Abstract - In this paper, an attempt has been made to show the important role of negative vortioity, that was introduced through theoretical considerations, by the analysis of actual data obtained of the " Kitty " and " Jane ". The fine structure of a tyhoon, that some small vortical cells of about 50-100 km in diameter and of signs of vorticity + and - are embedded in a large vortex, was elucidated by the following observational facts: the distributions of wind velocity and of horizontal convergence and divergence, the intensity of rainfall and the changes of pressure distribution with time. The schematic diagram of the vertical cross section of a typhoon is set forth, based on these new observational features. In addition to these problems, the oscillation of a vortical cell in a typhoon is discussed.

•˜ 1. Introduction

The theoretical studies of the structure of atmospheric vortices have been developed by many authors since the end of the 1.9 th century (for example, C. M. Guldberg and H. Mohn

(1876), A. Oberbeck (18118), D. Kitao (1887) and others) from purely mathematical stand points. But owing to the scarcity of appropriate data, it has been hardly possible to elucidate the actual structure of atmospheric vortices, especially of hurricanes and typhoons.

Recently, numerous aerological data at up to the heights 20 km have been obtained.

Based on these materials, studies of the atmospheric vortices have made a considerable progress, and some models of hurricane and have come to be discussed.

(for example, J. S. Sawyer (1949), C. E. Deppermann (1947) and H. Riehl (1950)). While one of the present authors (Syono (1948, 1951)) suggested the existence of a region of negative relative vorti city (anticyclonic v orticity) around' that of the positive one (cyclonic vorticity) as shown in Fig. 1. This region was at first introduced from a theoretical point of view, to avoid the divergence of the integrals of the kinetic energy and pre ssure difference which appear in using the RankinPs combined Fig. 1 Schematic distri vortex as a model of tropical cyclone. And the existence of bution of relative vorti this region was shown also by him from the wind velocity city in a typhoon

* Division of Meteorology, Contribution No. 35.

- 1- 398 Journ. Met. Soc. , Vol. 29, No. 12, 1951

distribution of Typhoon "Okinawa ". Such a model of cyclone was first noticed by Luighi

di Marchi (1893) from a different standpoint.

Afterwards Syono (1950) put forward the concept of " vortical rain " in, addition to the

convective, frontal and orographic rains. For a semi-infinite air column of unit cross sec

tion, the intensity of vortical rain is proportional to the horizontal mass convergence in it.

The horizontal mass convergence in the frictional layer is proportional to the vorticity of

gradient wind. Thus the intensity of rainfall is proportional to that of the vorticity in a typhoon. Since the region of negative relative vorticity is that of horizontal divergence,

we cannot expect any rainfall in it. This idea was verified by him using the data of

Typhoon " Okinawa ".

One of the authors (A. Kasahara (1950 a)) also studied theoretically the distribution

of the intensity of rainfall in Typhoon "Kitty". According to his previous study (A.

Kasahara (1949)), the speed of filling-up of a typhoon is also proportional to the horizontal

mass convergence. This result was supported by the agreement between the time change of

the pressure depth at the centre and the filling-up index in the case of Typhoon "Kitty"

(A. Kasahara (1950a)), This fact tells us that the filling-up does not take place in the whole

region of a typhoon (see •˜ 5).

Thus the horizontal mass convergence and divergence are very , important in explaining

the distribution of rainfall intensities and the filling-up as well as the relative vorticity

of a typhoon.

It is the purpose of this paper to show synoptically the physical meaning of the nega

tive vorticity which has been introduced from the purely theoretical standpoint and to

propose the schematic diagram of the structure of a typhoon along the line of above men tioned arguments.

Lastly, in addition to these problems, the oscillation of a vortical cell in a typhoon

is discussed. It would appear to be reasonable from this point of view that the vortical

cells are embedded , in a typhoon (see •˜ 6).

•˜ 2. Observations of negative vorticity in typhoons (I)

By synoptic studies of the vorticity distribution in typhoons, the existence of negative

vorticity regions has been made known by some authors as will be shown later. Its

important meaning, however, was not discussed in these studies owing to the lack of theory

concerning the negative vorticity in a typhoon.

(i) Typhoons which appeared during August-November, 1931

T. Ootani and H. Hatakeyama (1932) studied statistically the vorticity distribution in

typhoons which appeared in the neighbourhood of Japan during August-November in 1931,

using the data of upper air currents obtained by the pibal observations at 51 stations in

Japan. In that study they regarded that all observations made at the equal distance and

azimuth from the centres of typhoons are equivalent, disregarding the time of observations

and the position of every typhoon.

- 2 - On the Negative Vorticity in a Typhoon 399

Fig. 2 shows the vorticity* distributions at each level obtained from the distribution of wind velocity. Corresponding to these vorticity distri butions, we may expect that convergence is ac companied with positive vorticity, and divergence with negative. Fig. 3 shows the distribution of calculated horizontal convergence and divergence from the observed data of winds. In these figures,

Fig. 3 The distribution of calculated horizontal convergence and divergen ce (unit : m3 sec--1 km-2)

Fig. 4 The mean distribution of wind velocity in typhoons Fig. 2 The vorticity distributions at each level obtained from the dist ribution of wind velocity (unit : m3 sec-1 km-2)

* Hereafter, this term means the relative vorticity as far as not otherweise provided.

- 3 - 400 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951 we can see the alternate distribution of convergence and divergence as well as the negative and positive vorticities. Fig. 4 shows the distribution of calculated vertical wind velocity which is obtained under the assumption that the horizontal mass convergence below is canceled by the hori zontal mass divergence aloft, so that the state is maintained stationary. Within the distance 600 km from the centre, there exist ascending currents. On the other hand, there exist descending currents beyond 600 km. This fact may be a very remarkable feature concerning the structure of typhoons.

(ii) Typhoon "Muroto " (which attacked the Kansai district on Sept. 20, 1934)

T. Yamamoto and K. Takeda (Central Meteo rological Observatory (1935)) computed the vorticity distribution in Typhoon "Muroto " from the gra dient wind distribution at 500 in level. Fig. 5 shows the vorticity distribution. From this we may find also the region of negative vorticity around that of the positive one.

•˜ 3. Observation of negative vorticity in

typhoons (II) -General comments of negative vorticity - Fig. 5 The distribution of vorticity (i) The radial distribution of meteorological in Tyhoon Muroto (unit : 10-5 C. G. S.) elements in a typhoon It may seem reasonable to compute the distribution of meteorological elements, for example, wind velocity, pressure, etc. in a typhoon from the weather charts at each map time. We are puzzled, however, where we should choose the radial cross section of a, typhoon in order to get the representative distribution of meteorological elements, by reason. that isobars in a typhoon are scarcely circular As a rule, the time section based on meteorological elements which are observed at one station, is used instead of the above procedurefor the sake of convenience. But this time cross section of a typhoon may not be considered as a space cross section, since the moving speed of a typhoon is not uniform and the more remote the station is from the centre of typhoon, the more markedly this effect may arise. Then, in order to get a correct space cross section, it is necessary to rearrange the time axis of the time cross section to the axis of the radial distance of the station fromm the centre of typhoon because the distance may be measured on the weather chart. By this procedure it becomes possible not only to get the radial distribution of meteorological: elements, but also to compute that of horizontal convergence and divergence as well as that of vorticity in a typhoon. The change of successive space cross section obtained at each, station may be considered as the time change of the distribution of meteorological elements.

- 4 - On the Negative Vorticity in a Typhoon 401

In order to get the information about the distribution of meteorological elements near the centre of a typhoon, it is desirable to use the space cross section referred to the station, where the centre of the typhoon passed by . In this paper, only the space cross sections

ahead of the centre of a typhoon are analyzed.

The typhoons which are the objects of analysis hereafter are mainly Typhoons "Kitty" and "Jane ". The former attacked the Kanto district, the latter the Kansai district in

Japan, and they caused serious damages.

Typhoon "Kitty" appeared on August 27 , 1949 and was named "Kitty" at 9 a. M. on the 28 th. On the other hand, Typhoon "Jane" appeared on August 28, 1950 and

was named "Jane" on the 1st of September .

The tracks of these typhoons and the names and locations of meteorological stations

are shown in Fig. 6. The conspicuous features of these typhoons were that their circular

isobars were maintained and that no front appeared even after their landing. Fig. 7 shows

the surface map for 0600 P. M. T., September 8 , 1951 in the case of Typhoon " Jane ".

(ii) Vorticity distribution

In terms of cylindrical coordinates, the vertical component of vorticity of the gradient

wind •¬ is expressed by •¬

where r is the distance from the centre, ƒË3 the

Fig. 6 The tracks of Typhoons "Kitty" and " Jane " and the names and loca tions of the stations mentioned in this paper. The areas inside the circles in this figure show the mean positive vorticity regions used in the analysis Fig. 7 Surface map of 0600 P. M. T. of the filling-up September 3, 1950

- 5 - 402 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951

surface wind velocity and K the conversion coefficient from the surface wind velocity to the gradient wind. In the present case, we are considering a locally axial- symmetric typhoon.

K is given by •¬ where ƒµ is the angle between the isobar and the surface wind (S. Syono (1949)). For example,

Thus the vorticity of gradient wind is K times as large as that of surface wind.

The vorticity may be readily computed from the rv8-diagram by the graphical dif ferentiation using the relation (1). In Figs. 8 and 9 are shown the rv8-diagrams and the distributions of surface vorticity Ā8,, computed from the data of Typhoon " Kitty " obtained at the stations Oshima and Niizima.

The positive vorticity is concentrated, of course, near the centre as may be expected, and the region of negative vorticity exists in each case around that of positive one from the distance about 100 km outward. This fact will become important when we discuss the structure of a typhoon. Concerning the alternate appearance of the positive and negative vorticity regions, we will discuss in due order.

Figs. 8 and 9 show the distribution of calculated vorticity and observed intensity of rainfall and the rv,-diagrams at each station

- 6 - On the Negative VTorticity in a Typhoon 403

The necessity of the existence of the negative vorticity region was set forth by Syono

(1948, 1951) in his discussion on the problem of a model of atmospheric vortices as men tiqned in Ā1. We shall show another evidence of this fact as follows (A. ICasahara

(1950 b)) :

In terms of cylindrical coordinates, the vertical component of relative vorticity ƒÄ is expressed by •¬

where vs and yr are the tangential and radial components of the wind velocity respectively. Under the assumption of the wind velocity v=0 at r=0 (the centre) and r=R (the boundary of a typhoon), the result of integration of i; over the whole domain of a typhoon is

Then if the positive vorticity is concentrated in some finite regions, the region of negative vorticity.rmust exist around it. This relation holds for any arbitrary vorticity distribution, for example, as shown in Fig. 10.

Fig. 10 Schematic vorticity distribu tion; (a) cellular type. (b) annular type (Hatched regions indicate the areas of positive vorticity)

Then it is very natural for theoretical Fig. 11 Schematic distribution of vor ticity, the region inside ro is an treatment, to take the domain of a typhoon inner one and outside it is an outer as an area with the boundary v = 0. Owing to one. R denotes the boundary of the the lack of attention to the negative vorticity outer region. and because of its small order of magnitude, its important meaning has not yet come into question. For example, we assume the vorticity distribution as shown in Fig. 11, i. e. the value of the vorticity at the centre %to is 30 x 10-5 c. g. s. (actually, this was observed in the

- 7 - 404 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951

neighbourhood of the centre in Typhoon Jane) and decreased linearly from the value at

the centre to zero at r=ro (100 km).

Then the vorticity distribution in the inner region (Ām) is expressed by

ƒÄm= -30•~10-7,r+30•~10-5 c. g. s.

where the unit of r is km.

Using the relation •¬ where Sm is the area of the inner region and

Sont is that of the outer one, the radius of the outer region R is computed as follows:

where Āout is the relative vorticity in the outer region. Then for

ƒÄout=-0.4•~10-4, R=510km and for ƒÄout=-0.8•~10-4, R=367 km.

In the latter case, the vorticity in the outer region is equal to -2w sin P in middle

latitudes, so that this state is the limiting case for the above vorticity distribution in the

inner region according to the stability criterion obtained by the parcel method. The above

computed values of R are almost equal to the actual ones.

It seems to be much more plausible that the regions of positive and negative vorticity

exist alternately as shown in Fig. 10. Figs. 12, 13 and 14 are rv,-diagrams for Typhoon

Fig. 12-14 are rv,-diagrams at each station in the case of Typhoon "Jane". The hatched regions indicate that . of the negative vor ticity.

- 8 - On the Negative Vorticity'in a Typhoon 405

"Jane" . From these figures, we can easily find the position of the regions of negative vorticity (indicated by hatched lines). We may take 80 •` 100 km as the radius of the first inner region of positive vorticity (we call this region as the inner region) in this case.

•˜ 4. The rainfall *in a typhoon

(i) Vortical rain

It seems reasonable to think that rainfall in a typhoon may. be caused by its vortical nature, when no front appears. If we assume that the typhoon is locally axial-symmetric and is in a stationary state, the distribution of the intensity of rainfall W(r) in the typhoon, derived from the theory of vortical rain is expressed as follows (SySno 1950, 1951):

(2)

Here Y, is the vorticity at the ground, i; the mean vorticity in the frictional layer, A,=2w sin the mean absolute humidity, v the eddy viscosity coefficient, i the angle between the isobar and, the surface wind, and

Using the vorticity of surface wind i,(r) as denoted in •˜ 3, we can compute the

Fig. 15,16 and 17 the distribu tions of wind velocity, cal culated and observed intensity of rainfall in the case of Typhoon " Jane" ------wind velocity (m/s) - - - intensity of rainfall (Calculated) (mm/hour) ----- intensity of rainfall (observed) (mm/hour)

- 9 - 406 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1961

distribution of the intensity of vortical rain. Here we substitute ƒÄ•¬ fo ƒÄ in the formula

(2), and assume A is constant at each station and is so suitably determined that the com

puted distribution fits that of the observed.

In Figs. 1517, full lines show the observed surface wind velocity, and dashed lines,

the intensity of rainfall at each station computed by the formula (2), while dotted lines

show the observed intensity in Typhoon "Jane ". Computed distribution agrees with the

observed one at each station.

The value of A at each station is as follows:

Murotomisaki 8 •~ 102 c. g. s., SumotQ 8.29 •~ 102 c. g. s., 3 •~ 102 c. g. s.

These values are almost of the same order of magnitude as in Typhoon " Kitty ". (A.

Kasahara (1950 a))

(ii) Intermittent rain

From the discussions as given above, it is obvious that rainy regions may exist at

intervals of about 50100 km, as if they were in cellular or annular forms in the typhoon,

as shown schematically in Fig. 10. This is a remarkable fact and intermittent rain in a typhoon may arise when these regions pass through any stations.

For this intermittent rain, S. Fujiwhara and N. Yamade (1948) computed statistically

the time duration of rainfall T7 and of no rainfall TZ which are observed at one station

while typhoons pass through. The mean values of T7 and r2 given by them are about

1-2 hours. Then the horizontal dimension of the rainy region may be about 40~80 km,

provided that the moving velocity of the typhoon is about 50 km/hour.

In order to show the horizontal distribution of the intensity of rainfall and to give

the evidence of the existence of intermittent rain in a typhoon, it is better to draw

isohyets, using hourly observations: Fig. I8 shows the distribution of the intensity of rain

fall at every hour in Typhoon " Jane ". The numbers in these charts denote the intensity

of rainfall (mm/hour). These figures show more clearly that vortical cells are embedded

in any typhoon.

•˜ 5. Filling-up of a Typhoon a) Filling-upindex It is well known that a typhoon is filled up and its intensity decreases rapidly, after it landed from the sea. This mechanism was studied by one of the present authors (A. Kasahara, 1949, 1950a). The pressure tendency at a height z is given by the tendency equation,

(3)

In this equation the term (pw), is the sum of the vertical mass transport of air and that of precipitation, i.e. (pw),=(pairwair),+(pprecwprec)z. Of course, on the ground where z=0, (pairwair)o vanishes. But we can not neglect the effect of the term (pprecivprec)in the,

- 10 - On the Negative Vorticity in a Typhoon 407

Fig. 18 Distributions of the intensity of the rainfall in Typhoon - Jane" at every hour,

The time changes of its distribution suggest the cellular structure of typhoon, as shown in Fig. 10. The behaviours of 2 and 4 mm/hour isohyets are most attractive. The notation •~ denotes the centre of typhoon.

- 11 - 408 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951 pressure tendency (3), since when the intensity of rainfall is about 13 mm/hour it causes 1 mmHg/hour deepening in pressure. When the pressure tendency vanishes, one of the following two cases must hold. The one is that the horizontal mass convergence or divergence Q is zero throughout the air column, the other is that the horizontal mass convergence below z=H is canceled by the horizontal mass divergence in the upper layer above z=H as shown in Fig. 19, i. e.

Then the filling-up of a typhoon begins when the hori zontal mass convergence below z=H, caused mainly by the frictional effect of the ground, over comes the horizontal mass divergence in the upper layer above z=H, i. e.

If we neglect the horizontal mass divergence in the upper layers, the speed of filling

per unit time is proportional to the horizontal mass convergence in the lower layers. Then

the filling takes place in the horizontally convergent region (region of positive vorticity)

and its speed is proportional to the vorticity at that place. Then the pressure tendency at

the centre of a typhoon can be looked upon-as a good measure of the filling, since posi

tive vorticity may be almost concentrated in the vicinity of the centre of a typhoon.

To make clear the above argument quantitatively, it will be convenient to , examine the

filling-up index. We assume the region of the filling (region of positive vorticity) is

circular, and we divide it into four concentric parts as shown in Fig. 20. Then we mul

tiply each area of the land within the landed typhoon by the value of •¬ in each

annular region, •¬ , and ƒÉ being the relative vorticity of the surface wind and Coriolis

parameter respectively, and then we take summation of these four products. Since this

Fig. 20 The mean positive Fig. 21 The hourly change of pressure depth at the vorticity region (in the case centre (dotted line) and filling up index (full- line) with of Typhoon 'Jane") time from P. M. T. 0400 Sept. 3, 1950 (Typhoon "Jane ,") - 12 - On the Negative Vorticity in a Typhoon 409

sum is proportionalto the massconvergence , we callthis the filling-up index (A. Kasahara (1950 a)). In the case of Typhoon "Jane" , we may consider that the mean region of positive vorticity is a circle having the radius of 80 km and the values of vorticities in four parts

are 3.75, 11.25, 18.25, 26.25 X 10' c. g. s., respectively. In Fig. 21 the filling-up index at every hour is shown by the full line , and the obser ved pressure depth at the centre* is shown by the dotted line . The close agreement between

them in the initial stage of filling

up will support the above considera

tion.

b) The time change of pressure

distribution

For the purpose to show the

filling-up more clearly, it is better to

trace the time change of the pressure

distribution in a typhoon during its

travel.

Fig. 22 shows the pressure dist

ribution at each station in Typhoon

"Jane" using the same method as

we proposed when we got the wind

distribution in a typhoon as descri

bed in •˜3.

Figs. 23 and 24 are similar

figures as mentioned above in Typho

ons " Kitty " and "Muroto ". (The

names and locations of meteorological

stations are shown already in Fig. 6)

We can easily find a remarkable

feature that the filling-up takes place

inside the region having the radius

r and the deepening outside it.

The pressures at the points at the

distance r from the centre do not cha

nge. This radius r is found to agree

well with that of the region of positive

* Reports on the anomalous weather Fig. 24 Typhoon " Muroto " and climate, No. 9, 1950 (in Japanese) Cent. Meteor. Obs. Tokyo, Japan Fig. 22.24 The change in the pressure distribution

- 13 - 410 Journ. Met. Soc. Japan, Vol. 29, No. 12,1951 vorticity which was determined from the wind velocity distribution. We have really r = 80 km for Typhoons " Jane " and "Muroto ", and r=200 km for Typhoon "Kitty ". In order that a typhoon may keep its stationary state, the horizontal mass convergence in the inner region which is caused by the frictional effect of the ground must be com pensated by the- horizontal mass divergence in the upper layers, and on the other hand, the mass divergence near the ground in the outer region must be compensated by the mass convergence in the upper layers. From the above mentioned point of view, it is an easy matter to propose a schematic cross section in a typhoon as shown in Fig. 25. Concerning the horizontal mass divergence in the upper layers, C. E. Deppermann (1947) expressed as follows; "Possible divergence aloft especially arising from a fast upper current disposing of the rising air of the core ". In another paper of his, the presence of the region of descending air current around that of the intense ascending current is pointed out (Dep Fig. 25 Schematic cross section in a permann (1946)). A similar vertical cross typhoon. Arrows denote the current section is also stated by J. S. Sawyer (1947) system. Oblique lines show the rainy region. and H. Riehl (1950). The pressure tendency per unit time at the ground is shown as follows (A. Kasahara 1950 a):

(4) where B is the tendency coefficient and C denotes the value which gives the measure of filling, in another words, (1-C) X 1000 of the horizontal mass convergence is discharged by the horizontal mass divergence in the upper layers. Now we shall determine C from the actual data. The value of B may be given from the precipitation coefficient A by using the following transformation:

(5) where p and q denote the air density and the absolute humidity, respectively. Substituting A=3 •~ 102 c. g. s. and q=25/106 and ƒÏ=1.1•~10-3 c. g. s. into the formula (5), we get

•¬. If we assume •¬ c. g. s., then •¬

In the case of Typhoon "Jane". the observed rate of filling was about 2.4 mb/hour in the annular region having the radius 1060 km. From this we can decide C as follows: 2.4 mb=O 40 mb. Then C=0.06, i. e. 6% of the horizontal mass convergence contributes to

-14- On the Negative Vorticity in a Typhoon 411 the filling-up and 94% are evicted as the horizontal mass divergence in the upper layers. So we may estimate the velocity of ascending air current as follows:

40mb •~ 0.94 = 37.6 mb=gƒÏw. Then w=10,4 cm. It is plausible to consider that this is the mean ascending velocity in the vicinity of the centre of a typhoon. c) Consideration from the mass distributions and the structure of a typhoon . How does the mutual ex change of the air between the inner and outer regions take place ? From where is replenis hed the air that is contributed to the filling in the inner region ? Which mechanism, dynamical or thermodynamical, causes that the region of posi tive and negative vorticity papear alternately as expected by the rainfall distribution ? To answer these questions, it would be desirable to study the filling-up from the stand point of the time change of mass distribution. The total mass of the an nular region having the radius rl and rz is given by

where H is the proportional constant. For the sake of sim plicity, we plotted r (1000-pa) Fig. 26 The time changes of the relative mass dis instead of rpa as the ordinate tribution, In this figure, D and P denote the filling-up and deepening regions, respectively. and the distance r from the centre as the abscissa in a graph. Then we can readily see that the area presented in this figure indicates the relative mass in a typhoon. The relative mass distribution referred to each station in typhoon "Jane" is given in Fig. 26. From the time changes of the relative mass distribution, we can easily see shat the filling-up takes place in the vicinity of the centre and the deepening around it,, as expected

- 15 - 412 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951 in (b). But this tendency is found in the outer region and the filling-up and deepening tendencies are represented by the notatinos D and F respectively in this figure. Thus the alternate distribution of the region of horizontal convergence and divergence, i. e. the region of positive and negative vorticities are also pressented independently as mentioned in §4. From the comparison of the successive distribution referred to each station, it seems reasonable to expect that during shorter intervals the total mass increased in the inner region (the radius r 80 km) would be equal to the decreased in the outer one. Along this line of argument, a schema- tic cross section of vertical circulation in a typhoon may be slightly modified as shown in Fig. 27. This structure of typhoon is frequently observed by aircrafts-recently. (Cent. Met. Obs. (1950)). The mechanism of generation of the alternate distribution of the vorticities as Fig. 27 Schematic cross section in a typhoon. (Modified) well as the divergence and convergence will be studied in the near future.

§ 6. Oscillation of a vortical cell* On the time cross section with the wind velocity, there appears the oscillation of wind velocity with periods about 1~2 hours as shown in Fig. 28. This may be due to the cellular structure of a typhoon as men- tioned in § 3. In this section, we intend to discuss the oscillation of a vortical cell in a typhoon. By solving the perturbation equations Fig. 28 Pressure tube anemograph record concerning the perturbed pressure p super- obtained at Kobe (Sept. 3, 1950 Typho- imposed on a steady basic pressure field on " Jane ") which may be deduced from the equations of motion (three components), the equation of continuity and the equation from the condition that the motion be adiabatic, it maybe shown that the perturbations with velocity components v,•, vei v,, pressure p and densityfp are proportional to , e aD XJB(kr) and the period of oscillation of this perturbation system is

* The detailed discussion of this oscillation of a vortex will be published By S. SyOno in a paper titled "On the theory of formation of tropical cyclones" in the near future.

-16- On the Negative Vorticity in a Typhoon 413

(7)*

where •¬ (Z: absolute vorticity in a basic field, io: tan gential wind velocity in a basic field, O: potential temperature, g: acceleraion of gravity, D: vertical scale of disturbances, 1(k: horizontal scale of disturbances, and v: frequency). When the horizontal scale of disturbance is considerably larger than the vertical one

This is the period due to the inertia oscillation o f a vortex and is of the order of magnitude of several hours (H. Arakawa (1940), K. Gambo (1949)). On the contrary, when the horizontal scale of disturbance is considerably smaller than the vertical one (1/k <

into the formula (7), we get

Thus the period of oscillation of a vertical cell may be about 1-2 hours.

The value of k may be estimated from the distance between the centre and a point

where the perturbed wind velocity and pressure will vanish, i. e. Jo(kr) =0. The roots of

Jo(ƒÌ) =0 are ƒÌ=2.4, 5.5, 8.6, 11.7 and 14.9, etc., then, if we take the radius of the first

inner region of positive vorticity to be 80 km, the value of 1/k will be estimated from the

first root of Jo(kr)=0. Then

The second positive region will appear from 183 to 287 km corresponding to the second

and third roots of Jo(ƒÌ) =0. The third positive region will appear from 390 to 500 km.

Of course, it must call attention that this is only one component of oscillation about k in

Even if we assume that the disturbances in a typhoon are expressed by

* This representation may be derived from the similar perturbation equation which was discussed in the Cartesian co-ordinates by K. Gambo in a paper 11On the dynamics of planetary waves and the role of the Richardson number " (to be published). -17- 414 Journ. Met. Soc. Japan, Vol. 29, No. 12, 1951

(axial-antisymmetric disturbances), there is no essntial difference as far as the period of oscillation between the cases of axial-symmetric-and -antisymmetric disturbances, provided that the vertical wind shear is very small.

Acknowledgments : -The authors wish to express their hearty thanks to the Central

Meteorological Obserbatory for providing facilities to use the data of Typhoons "Jane"

and "Kitty" and to the Science-Research Council of Ministry of Education for its kind

financial aid to this work. We are also indebted to Miss. M. Okubo for her valuable

assistance.

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