On the Motion of a Typhoon (I)*

Transcription

1 On the Motion of a Typhoon (I)* By S. Syono Geophysical Institute, Tokyo University (Manuscript received 2 November 1955) Abstract Solving barotropic vorticity equation, the motion of a disturbance of typhoon-type is treated. First, solving barotropic vorticity equation, the deformation of an initially circular disturbance in a changing basic flow with constant shear is treated. Second, the velocity components are given by U+Eyo+B/C, V, where U and V are uniform basic flow, is constant shear of the basic field, yo is the northward component of the coordinates of the centre, p is the latitudinal variation of Coriolis factor, C is a negative constant concerned with the scale of the disturbance. Third, equations of motion of the disturbance are obtained. Fifth, the calculated periods and amplitudes of meandering motion are compared with those of actual typhoons. 1. Introduction mass of the typhoon of unit slice, the density The motion of atmospheric vortices has and cyclic constant, the Coriolis force is been treated by many authors1)-10). However, neglected. We can put pk=m, where C is the results are diverse. Kuo4), Yeh9) and the mean vorticity in the domain of the Takeuchi8) adopted solid circular cylinder typhoon. Then we get T=. The models and Rossby5), Yoshitake10) and the period is independent of the radius of the present author') adopted fluid mass models. vortex, then this formula may be applied to Bjerknes and Holmboe treated circular pressure the vortex filament, i.e. the vortex filament pattern1). has a proper period of oscillation. This conclusion According to the universally accepted classical contradicts to the result of the classi- theory on the motion of a vortex filament, theory. Further, if pk is constant, that a vortex filament moves with the velocity of meansthe intensity of the typhoon does not the basic flow. Further according to experiences change, according to his formula T increases on recent numerical prediction, we know with increase of latitude. But in actual that the vorticity at a mass point is driven cases the period of meandering motion has a by the flow caused by the surrounding vorticity clear tendency that it decreases with increase field, that is considered as the basic flow of latitude. From the above consideration, at the point. The movement of a cyclone is the solid circular cylinder model, which accompanies predicted by vorticity advection and vorticity horizontal escaped motion, additive change due to Rossby effect, provided that mass and rotor effect, seems to be unnatural. barotropic atmosphere is assumed. No other The models proposed by the present force acts to the vortex. So far as lower author and Yoshitake, are not solid body convergence and upper divergence are neglected, models. But they treated rotating fluid mass a typhoon is nothing but a fluid mass within finite area and obtained equations of with concentrated vorticity. When the radius motion. The motion, which is obtained by of the typhoon tends to zero, the motion should solving the equations of motion, is not equal coincide with that of a vortex filament given to the motion of the surrounding fluid. Then by the classical theory, provided that the effect there must be some kind of escaped motion, of Coriolis force is neglected. According to which causes complicated additive pressure Yeh's result, the period of oscillation is given additive velocity. The present author by T=4irM(pK)-1, where M, p and K are the neglected tacitly this term under the assumption * Division of meteorology, Contribution No. 85. that it is not large. Yoshitake, also, did not touch upon this problem. Now, it seems to the present author that the deformable -13-

2 246 Jour. Met. Soc. Japan, Vol. 33, No rotating fluid mass model is not profitable, because it is not easy to eliminate rationally certain theoretical ambiguity. Sasaki and Miyakoda ) treated the barotropic vorticity equation and obtained the velocity of the centre of a typhoon in the general field with shear. They developed a method for the numerical prediction of the trajectory of a typhoon and after then experiments of numerical prediction of the trajectory of a typhoon has been carried on by staff members of Numerical Prediction Research Group in Tokyo. They obtained only the velocity of the centre of a typhoon. However, their approach seems to be hopeful for development of the theory of motion of a typhoon. Most of authors who treated the meandering motion of typhoon including the present author, thought that the meandering motion is the proper motion of atmospheric vortices. But due to the difficulty of interpretation stated above, the present author had the question about the proper motion and inclined to think that the motion is that of the basic flow. For a while, he could not derive the meandering motion from the vorticity equation. But putting down the geostrophic assumption for the general flow, the meandering motion is derived. Independently, H. Fuchi had the same idea and investigated the meandering motion. He could explain the deviation of the period of meandering motion from that of the inertia period. In the present paper, the barotropic model is assumed as a first approximation. This assumption is good approximation in lower latitudes. But in middle latitudes, especially in autumn, the baroclinicity cannot be neglected. The motion of a typhoon in a baroclinic atmosphere is left in future investigation. First solving the barotropic vorticity equation, we treat the deformation of an initially circular vortical disturbance. Second, the velocity of the disturbance is obtained. Third, using the formula for the velocity, equations of motion of the disturbance are obtained. Fourth, solving the equations of motion, trajectories of the disturbances in a changing basic flow with constant shear. Fifth, the meandering motion for actual typhoons are investigated and comparison with the theoretical results are made. 2. Solution of Vorticity Equation We shall consider a two-dimentional inviscid atmosphere, where the curvature of the earth surface is neglected. The equations of motion are given by where x and y are cartesian coordinates, positive towards the east and towards the north respectively, p and a are the pressure and the specific volume of the air respectively. f and B are the Coriolis factor and its meridional gradient respectively. By taking cross differentiation, we get the vorticity equation (1) (2) where C and Q are the vertical component of the relative vorticity and the horizontal divergence respectively. The function J is the Jacobian, that denotes (3) We assume that Q and J(p, a) vanish. Then the vorticity equation becomes (4) In this case, we can express the velocity components and the vorticity by using stream function T, i.e. Now we assume that the stream function is given by (5) (6) where U and V are arbitrary functions of time, E is an arbitrary constant and (P is the stream function for a disturbance. Then (4) becomes (7) where j3v is dropped, because this terms remains when 0 vanishes. This term should be dropped by the same base of the assump. tion of non-divergence. We shall treat the motion of a revolving disturbance, which has almost circular pattern. Then we put (8) where 0, and 4,n represent circular and noncircular patterns -14-

3 On the Motion of a Typhoon (1) 247 Putting (8) into (7), we get (9) (19) so that Now we assume (10) 11 (20) into (18), we get (20) where xo and yo are functions of time. C is a negative constant. and (11) into (10), we get Now we put Then (16) into (14), we get Further we put then (12) (13) (14) (15) (16) (17) (18) where (21) (22) The equation (21) gives the timely change in the non-circular part of the moving disturbance. Solving (21), we get the deformation of the shape of the disturbance. Now we consider the meaning of the right hand side of (21). The first term gives the advective change in the vorticity of the circular part by the shearing motion of the basic flow. The second term gives the advective change in the vorticity of the non-circular part by the shearing motion of the basic flow. The third term gives the vorticity change in the non circular part due to the latitude effect. The fourth term gives the advective change in non-circular part due to the circular motion. The fifth term gives the advective change of non-circular part by non-circular motion. 3. Deformation of the disturbance Now we consider the deformation of the circular disturbance. We assume that the initial disturbance is given by (11), that means vanishes at t=0. Now we expand lv into power series of t; (23) (23) into (21), we get 15

4 248 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 (24) Putting coefficients of each power of t equal to zero, we get (25) (26) (27) First we consider (25). Rewriting (25), we get Taking account of (12), (28) becomes (28) (29) Integrating spatially (30) Here an integrating function A on the right hand side (30) is neglected, because it satisfies A=0 and vanishes on a sufficiently large circle surrounding the disturbance, then it vanishes everywhere. Next consider (26). (25) and (30) into (26), we get Using (12), we get (31) Putting (32) and introducing (32) and (33) into (31), we get (33) (34) Next we shall obtain. Differentiating (11) with respect to X and Y, we get -16-

5 On the Motion of a Typhoon (1) 249 (35)-(38) into (33), we get (40) Now we put (41) (41) into (40), we get -17-

6 250 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 (42) Putting and introducing (43) into (42), we get (43) (44) If we put the third term vanishes and (44) becomes (45) Then we get (46) Integrating again (47) Then we get (48) G must be finite everywhere, then C1=CL=0. We obtain the same result by putting a=2 or-2, i.e. (49) (41) into (49), we get (50) (51) (49), (50) and (51) into (41), we get (52) -18-

7 On the Motion of a Typhoon (I) 251 (36), (38), (39) and (52) into (30) and (34), we get (53) Similarly, we can obtain higher terms of t. But tracing of deformation is not our main purpose, so that we stop at the second order terms with respect to t. In Fig. 1, the pattern is shown for the case, where following assumptions are adopted. 4. The velocity of disturbance The pattern F, (11) is circular and the coordinates of the centre are given by (15). The velocity components of the centre are given by (54) The total pattern is given by If the centre of does not move or moves very slowly on the moving coordinates with the velocity (54), the total pattern moves with the velocity (54). The velocity components of the centre of the pattern FC-~ Fn are given by expansions: Fig. 1. Deformation of stream lines. (a) The pattern of initial stream lines. (b) Radial distribution of initial stream function and tangential velocity. (c)-(f) Patterns of stream lines. (56) (55) Now we shall obtain Cx and CY. When r is very small, we have following -19-

8 252 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 (56) into (53), we get (57) Differentiating (11) and (57) with respect to X and Y and putting X=Y=0, we get (58) (58) into (55), we get (59) Then the centre of the pattern moves northwards with the velocity given by (59). Now we assume the values given in 3 Then Cr is negligibly small. Thus actually the pattern moves with the velocity given by (54). In (54), Q/C is additive velocity to the velocity of the basic flow. The amount of B/C is not large, but it is of cumulative nature, so that the effect gives systematic westward deviation from the track obtained by the basic flow only. In Table 1, 1/C in km/day is given for various latitudes and radia. The radia are defined by the first root of Bessel function of zero order i.e. ro=2.4( )-1. From Table 1, we see that for disturbance, whose radia defined as above are less than 600 km, the Rossby effect seems to be negligibly small, in the instantaneous velocity. But this effect is cumulative, so that it cannot be neglected for long range precise prediction of tracks of tropical cyclones. For instance, for the disturbance with radius 400 km, the westward deviation from the tracks obtained by the basic flow is 53 km in one day and 371 km in one week. 5. Equations of Motion of the disturbance Here we shall obtain equations of motion of the disturbance. The velocity components of the centre of the disturbance are given by (54). For the sake of mathematical simplicity, we assume & is constant. Putting and introducing X60) into (1), we get (60) Table 1. (km/day) where (61) -20-

9 On the Motion of a Typhoon (I) 253 and p is the pressure for the basic flow, and has the nature that does not decrease with increase of radial distance from their centre. On the other hand, p' is the pressure for the disturbance and has the nature that decreases with increase of radial distance from the centre. The left hand side of (62) has the nature that does not decrease with increase of radial distance from the centre. The right hand side of (62) decreases with increase of radial distance and tends to zero with indefinitely increase of r. Operating cross differentiation to (62), we get vorticity equation (7). is the solution of the vorticity equation, then if we put into the right hand side of (61), the right hand side can be put From (68), we get (69) (70) Strictly speaking, f is a function of y. But because f is a slowly varying function of y, we assume f is locally constant. Solving (67) and (69), we get U and V and introducing U and V into (54), we get the instantaneous velocity of the centre of the disturbance. Integrating the instantaneous velocity with respect to time, we get the trajectory of the centre of the disturbance. Now we shall obtain the equations of motion of the disturbance. Differentiating get (54) with respect to time, we (71) (72) (54), (67) (69) and (70) into (71) and (72), we get (63) where 0 is a certain function. Then if we take the right hand side of (61) vanish. K satisfies 2K= 0 and K vanishes at a sufficiently large r, then K=O. Then equations (62) become (64) (65) p changes with the change in the motion of basic flow, which is governed by the vorticity equation. But so far as the motion of the disturbance concerned, we can assume p is a given function. Now we assume following form as p: (66) where A and D are arbitrary functions of time and B is constant. This form is appropriate to represent the pressure pattern in the western part of Pacific anticyclone. (66) and (62) into (65), we get (67) (68) (73) (74) These equations are the equations of motion of the centre of the disturbance and are similar to those obtained by the author elsewhere(7). 6. Trajectory of the disturbance Now we shall obtain the trajectory of the centre of the disturbance. Differentiating (74) with respect to time, we get (75) Eliminatin from (73) and (75), we get where (76) (77) Integrating (76) with respect to time, we get (78) The general solution of (78) is given by -21-

10 254 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 (79) Subtracting (78) from (74), and deviding by f, we get (80) Integrating with respect to t, we get (81) (79) into (81), we get (82) (78) and (82) are general solutions for xo and yo. Now we consider two examples. I. Aperiodically changing pressure field. The initial conditions mean that thedisturbance is moving uniformly with the gradient wind. These conditions may be interpreted that the disturbance is suddenly formed in the basic field. (83) (84) Then (82) and (79) become as follows: (85) (86) following formula (87) into (85) and (86), we get (88) -22-

11 On the Motion of a Typhoon (I) 255 Further we have following formulae: (89) (90) (90) into (88) and (89), we get (91) The velocity components are given by (92) (93) (94) initial conditions: (95) Then the solutions are given by (96) From the above solution, we can see that the motion is composed of the oscillatory motion (97) with period (98) and non-oscillatory motion. is about one tenth of f or smaller, then we can put. Then (96) and (97) reduce to -23-

12 256 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 (99) (100) First we consider the case of stationary pressure pattern. If the pressure pattern is stationary, i.e., (99) and (100) reduce to (101) (102) The motion is composed of non-oscillatory motion and oscillatory motion: (103) (104) Eliminating t from (103) and considering the relations between a, and Ao, Do, B, we get (105) (106) (105) is the equation for a parabola. The vertex (point of recurvature) is given by (107) On the other hand, the vertex of isobars are given by (108) The deviation of the point of recurvature from the vertex of the isobar, which passes the origin, depends on the scale of the disturbance. Now we consider the relation between the deviation and the scale of the disturbance. To show numerical example, we assume following numerical values: above values into (107), we get In Fig. 2, non-oscillatory trajectories of disturbances, whose scales are N=1, 2, 3, 4, 5, are shown. Broken lines are curves, which connects points of recurvature. Next we consider the oscillatory motion. For stationary pressure field, the oscillatory -24-

13 On the Motion of a Typhoon (I) 257 Fig. 2. Non-oscillatory trajectories of typhoons in a stationary parabolic field are shown by full lines. Curves, which connects the points of recurvature are shown by broken lines. N means the scale of typhoons defined by the first root of Bessel function of zero order. ro is the radius, at which 0, vanishes. for non-stationary pressure pattern. The pressure change is expressed by 8.1 and 8, and d,l are included in the same parentheses with d,zo, do and /3/C. The nature of the motion is already studied for the case of stationary pressure pattern. Now we shall compare the magnitudes of 8.1 and 8,1 with those of 8,o, 6,o and QIC. If the pressure gradient in x-direction changes one N.-th of the initial pressure gradient in one day, Fig. 3. Amplitudes of meandering motion of typhoon. Abscissa: latitude of the centre of typhoon, Ordinate: amplitude of meandering motion of typhoon. motion is caused by the curvature of the pressure pattern and the Rossby effect. In Fig. 3, the relation between amplitudes and the latitude is shown, where the abscissa is taken in latitude, the ordinate is taken in km. The latter oscillatory motion is more predominant except in latitudes lower than 10 N. In Fig. 4, calculated trajectory is shown, where the mean motion and oscillatory motion are taken account for the case N=3. Next we shall consider the oscillatory motion If the y-coordinate of the vertex of the basic pressure pattern displaces N, x 100 km in one day, Then N, N, and N are numbers less than ten. Then the first terms and second terms are -25-

14 258 Jour. Met. Soc. Japan, Vol. 33, No. 6, 1955 Fig. 4. Calculated trajectory of a typhoon in a stationary parabolic field. Smooth curve is the mean trajectory, meandering curve is the calculated trajection of the typhoon. The scale of typhoon is N=3. of comparable order of magnitude. Therefore, we can see that the change in pressure gradient causes the actual oscillatory motion of the disturbance. II. Periodically changing pressure field In the first example, we have seen that, oscillatory motion appear in a linearly changing pressure field and the period of the motions are that of inertia period, As the second example we shall study the motions in a periodically changing pressure field. Now we assume (109) (110) (109) into (81) and, (79), we get (111) Using (90) and following formulae (112) (113) - 26-

15 On the Motion of a Typhoon (I) 259 (111) and (112) reduce to (114) (115) Further using (40) and following formulae (116) (114) and (115) reduce to (117) (118) The velocity components are given by (119) (120) initial conditions (110), we get (121) (121) into (117) and (118), we get the solutions: (122) (123)

16 260 Jour. Met. Sac. Japan, Vol. 33, No. 6, 1955 If we put f2= p2, (122) and (123) reduce to (124) (125) The trajectory is composed of four different types of motion, i.e. parabolic motion, oscillatory motions with period and When q, and q2 are nearly equal to f, the amplitudes of motion due to the periodic change in pressure pattern become very large. q, and q2 are equal to f, resonance phenomenon occurs. For the case q1f and, q2f, (124) and (125) become (126) (127) From the above result, we see that oscillatory motion is excited by the periodic change in the pressure pattern. However, when the period of the latter tends to the inertia period, the amplitudes of the oscillatory motion increase and when the period becomes equal to the inertia period, the amplitudes increase with increase of time. Therefore, we may conclude that the inertia oscillation predominates in the actual trajectories of tropical cyclones. 7. Meandering Motion of typhoons (steering principle) and of that due to Rossby effect. The basic flow oscillates around the mean flow by deviations from geostrophic flow. The period of the oscillatory motion is that of inertia oscillation. Even in the oscillatory pressure field, the amplitude of the oscillatory motion is most predominant, when the period of the basic pressure flow is equal to that of inertia oscillation. In actual cases, the pressure pattern is different from one here assumed and changes in more irregular manner, therefore actual trajectories may be more complicated than here obtained. However, when the pressure pattern is not and does not too much complicated, solutions obtained above represent approximate feature of actual trajectories. Therefore in the actual trajectory of a typhoon, the inertia oscillation type may be most probable. The amplitudes of the oscillation decrease in proportion to the inverse square or cubic of sine of latitude. We shall call the motion as the meandering motion of a typhoon. In Fig. 5, the relation between In the former sections, we have seen that the disturbance of typhoon-type moves with the resultant velocity of that of the basic flow Fig. 5. Relation between period of meandering motion and latitude. Ordinate : period, abscissa : latitude. Solid line is curve for inertial oscillation

17 On the Motion of a Typhoon (I) 261 the period of the meandering motion and the sine of latitude is shown for actual typhoons in 1950 and The solid line is that of inertia oscillation. We can see that the period of meandering motion is almost that of inertia oscillation. H. Fuchi investigated the period of meandering motion and he made clear the cause of deviation from the inertia period. In Fig. 6, the relation between the amplitude and square of sine of latitude is shown*. The location of typhoons were determined by map analysis and reports of aircraft weather reconaissance flight carried on by U.S. airforce at Guam Island. Fig. 6. Relation between amplitude of meandering motion and square of sine of latitude. Ordinate: amplitude in km, abscissa : inverse square of sine of latitude. References 1. Bjerknes, J. and Holmboe, J. (1948) : On the theory of cyclones J. Meteor. 1, Fuchi, H. (1955) : On the oscillatory inertia motion of the atmosphere and the meandering movement of typhoons. Geophys. Mag. (in print) 3. James, R. W. (1952) : Note on the motion of atmospheric vortices J. Meteor Kuo, J. (1950): The atmospheric vortices and the general circulation J. Meteor. 7, Rossby, C. G. (1948): On the displacement and intensity changes of atmospheric vortices. J. marine Res. 5, (1949): On the Mechanism for the release of potential energy in the atmosphere. J. Meteor. 6, Sasaki, Y. and Miyakoda, K. (1954): Numerical Forecasting of the Movement of Cyclone. Collect. Met. Pap. Tokyo Univ. 5, No. 74. Sasaki, Y. (1955): Barotropic Forecasting for Displacement of Typhoon. Collect. Met. Pap. Tokyo Univ. 5, No Syono, S. (1951): On the Motion of a Vortex in a Non-Uniform Pressure Field Pap. in Met. Geophys. 2, Takeuchi, H. (1953) : The Motion of Tropical Cyclones in a Non-Uniform Flow Field. Pap. in Met. Geophys. 3, Yeh, T. C. (1950): The motion of tropical storms under the influence of a superposed southerly current. J. Meteor. 7, Yoshitake, M. (1953): On the dynamics of Open Systems and its Application to the Motion of Typhoons. Geophys. Mag. 24, Acknowledgement: The present author wishes to express his hearty thanks to Miss M. Okubo for her help in numerical calculation. * These figures are made by M. lida and T. Watanabe by author's suggestion. -29-