On the Small Perturbations Superposing Upon the Circular Vortex

551. 515. On the Small Perturbations Superposing upon the Circular Vortex by Y. Masuda Meteorological Research Institute (Received. April 1, 1952) Abstract Mainly, from the analysis of vorticity in the case of the typhoon, it is found that small perturbations superpose upon the circular vortex seemingly uniform at a glance, and that they travel as wave-motions. Furthermore, as an attempt to explain this phenomenon, approximate solutions of the perturbation equation of the circular vortex whose basic currents are gradient winds, are obtained, and this theoretical result is com pared with the analysis. Introduction In a region of an intense vortex like a typhoon, periodic rainfalls are observed frequently, and the study of this phenomenon was first taken up by S. FUJIWHARA and N. YAMADA [1]. S. SYONO [2] investigated the relation between rain and vorticity in a typhoon region and named it vortical rain. A. KASAHARA[3] and T. ANDO [4] discussed the same phenomenon individually, and showed the discon- tinuous distribution of rainfall intensity in a typhoon region. In winter a'so, when the Siberian Anticyclone spreads out towards Japan, it is well known that a remarkable periodicity is seen in the precipitation in the Japanese districts along the Japan Sea. Therefore, we may think that, though the intense cyclonic or anticy- clonic vortex such as the typhoon or the Siberian Anticyclone seems to be uniform at a glance, small perturbations superpose upon it. In this report, using the surface meteorological elements such as pressure, precipitation and vorticity etc. in a typhoon region, the existence of small perturbations superposing upon the typhoon i,s shown and the question whether these perturbations have a character of the wave-motion or not is solved. Further, on the assumption of the gradient wind of the circular isobars, the perturbation equation is solved and the displacement velocities of these perturbations are deduced. Part I Analysis of small perturbations 1. Distributions of meteorological elements in a typhoon area a) Pressure distributions Figs. I and 2 show the variations of pressure tendencies (2p/t2) for each one hour for the typhoons Jane (Sep. 3, 1950) and Kezia (Sep. 13, 1950) respectively. Fig. 1. The variations of pressure tendencies (a'p/at2) for the typhoon Jane (0500-0700, Sep. 3, 1950). In this -figure, the hatched areas represent the areas of negative values of a2p/at2 and solid lines are drawn for each 2 mb/hour2. In these figures, hatched areas represent the areas of negative values of -a2prai2, and solid lines are drawn for each 2 mb/hour2. From these figures, it is found that the positive and negative areas of a2pial2 are distributed alternately and vary with the time, except in the neighbourhood of the center of the typhoon. If the typhoon is a, uniform isobaric system, the distribution of a2plai2 must be almost symmetrical with respect to the center in the case of its steady movement. How- ever, as these figures show, the positive and negative areas of 3275laz2are distributed alternately and their signs change almost inverse- ly with the time. So we may suggest that Fig. 2. The variations of pressure ten a certain kind of perturbations superposes dencies (a2piat2) for the typhoon Kezia. upon the typhoon syStem. However, we can (1000-1200, Sep. 13, 1950). The ex- not, from this analysis only, deduce that planations of representation are same as in Fig. 1. these perturbations are wave motions. Near the typhoon center, the pressure variations are too large for the above perturbations to be detected. b) Distributions of precipitations As remarkable variations of precipitations in the typhoon region and their remarkable periodicities are well known from many analyses [1] [3] [4], we may omit the study of distributions of precipitations in the typhoon in this report, c) Vorticity distributions From the analysis of a215/at2, the fact that some perturbations superpose upon the typhoon system on the outer region about 300 km distance from the center of the typhoon and move with the time, is shown. In an intense vortex like the typhoon, isobars near the typhoon center are almost circular and very crowded, and seem to be almost uniform at a glance within about 200 km distance from the center. Therefore, it is difficult to detect the perturbations near the typhoon center from the isobars themselves. So in this report, in order to detect them easily, vorticity distributions are used. Vorticities are estimated by the radial distribution of the gradient winds which are computed by the next equation on the basis of the surface isobars, where r is, the distance from the typhoon center, co the angular velc city of the earth rotation, p the ap latitude , and -- the arp radial component of the pressure gradient. Prac- tically, vorticities are estimated on the intersecting points of the concentric circles whose center is the typhoon center and whose radii are the lengths of 0.6°, 0.8°, 1.0°, 1.2°, 1.4°, 1.6°, 1.8° and 2.0° in latitude respectively, and the lines which divide the whole circle by each 20 degrees. However, as the tangential components of winds are neglected in this method, vorticities estimated in this report do not represent real vorticities, but only the approximate aspects. Fig. 3 shows the vorticity distributions obtained from the above procedure in the typhoon Jane. In this figure, the hatched areas represent the areas of negative vorticities, and the solid lines represent the equi-vorticity lines for each 4 x 10-5 sec-1. Gene- rally, up to the present Fig. 3. The vorticity distributions in the typhoon time, it has been thought jar2le (0900-4200, Sep. 3, 1950) . In this figure, that vorticities near the the hatched areas show the areas of negative vorticities, the solid lines the equi-vorticity lines typhoon center are positive for each 4 x 10-5 sec-1, and the mark ® the ty- everywhere. So it is very phoon center. interesting that, as Fig. - Fig. 5. Deviations -for 11h and 12 h from mean state 3 shows, the areas of the negative vorticity are widely scattered, except in the immediate neighbourhood of the center. And further, the fact is remarkable that the areas of large and small vorticities distribute almost in a ring shape. In this report, mainly from the analysts of this vorticity distribution, the perturbations superposing upon the typhoon system are examined. 2. The inspection of the perturbation a) The mean state The mean values of the vorticities for each hour, which are computed over each concentric circle around the center, are shown in Table 1. In this table, as the rear of the typhoon is out- side of the domain of the weather map at 9 h and 10 h and so vorticities at that region can not be computed, the values for 9 h and 10 h are not so relia- ble. Therefore, the aver- age of the mean values for 11 h and 12 h is taken as the mean state. This value is shown in the lowest column in Table 1. Fig. 4 represents these mean values in Table 1. In this figure, though the vorticity of the mean state decreases steeply with the distance from the center, it decreases not uniformly but reaches its minimum at the distance 1.2° Lat. from the center and then increases a little and after then decreases. From the above fact, we may think that, though the vorticity inside the area of the radius 2.0° Lat. from the center is generally positive, the areas of large and small vorticities distribute in a ring shape. b) The perturbation of the radial direction If we assume that the small perturbations in the typhoon area are composed of perturbations of the radial and the tangential direction, only the perturbations of the radial direction are considered to re- main in the mean values of vorticity for each hour shown in Table 1. There- fore, we may think that the deviations from the mean state computed in the lowest column in Table 1 represent the perturbations of the radial direction. Fig. •5 •shows these deviations for 11 h and 12h: respectively. From this', figure, though the deviations for 11 h are negative,and those Fig. 6. The distributions of perturbed vorticity in the . for 12 h are' positiVe every- typhoon Jane (0900-1200, Sep. 3, 1950). In this figure, the hatched areas represent the areas of where, at the positions negative perturbed vorticity, the solid lines the where the deviations for equivorticity lines for each 4 x 10-5 sec- and the mark ® the typhoon center. 11 h are relatively large, 146 Y. MasudaVol. III No. 2 the deviations for 12h are relatively small. That is to say, the perturbation of the radial direction seems to be the standing wave having the wave length of about 0.4° in latitude. However, as thd cases analysed in this section are only two, the above result may not be a universal phenomenon. c) The perturbation of the tangential direction In order to obtain the perturbation of the tangential direction, deviations from the mean values of vorticity for each hour in Table 1 are taken. They are regarded as perturbed vorticities and their distributions are examined. Fig. 6 shows such distributions of perturbed vorticity obtained by the above method. In this figure batched areas represent the areas of the negative perturbed vorticity and solid lines are drawn for each ,4x10-5 sec-1.

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