A Further Study of Dynamics of the Four-Day Circulation. By Yoshihisa Matsuda

Transcription

1 February 1982 Y. Matsuda 245 A Further Study of Dynamics of the Four-Day Circulation in the Venus Atmosphere By Yoshihisa Matsuda Department of Astronomy and Earth Sciences, Tokyo Gakugei University, Koganei 184 (Manuscript received 31 July 1981, in revised form 5 November 1981) Abstract A steady state model on the four-day circulation in the Venus atmosphere proposed by Matsuda (1980) with use of two-layer model is examined by time integrations of a fivelayer model. By assuming infinite horizontal viscosity, it is shown that the fast zonal flow is gradually formed in the upper layer over 1,000 days by the accumulation of angular momentum. This angular momentum is supplied to the lower atmosphere from a slowly rotating planet and is transported upward by meridional circulation. Stationary solutions are obtained for various values of three external parameters (i.e., horizontal diffusion time, latitudinal differential heating and planetary speed of rotation). Multiple equilibrium states appear in the system when the horizontal diffusion time (normalized by the vertical one) 8 is reduced to The super-rotation rate as large as the observed one is realized in some solutions when 8 is reduced to The parameter dependence of (multiple) equilibrium states in the present model is analogous to that obtained by the two-layer model in Matsuda (1980). By comparison between the equilibrium states obtained in these two models, the prediction on the stability of multiple equilibrium states (Matsuda (1980)) is verified; the state with the fast zonal flow and the state with the strong meridional cell are stable and the state having the characteristics intermediate between these two states is unstable. Finally, by assuming two different ways of external heating (sudden heating and gradual heating), it is examined which of the two stable states is attained as a final state; the sudden heating leads to the state with strong meridional cell, while the gradual heating to the state with fast zonal flow. 1. Introduction Since the discovery of the fast zonal flow in the Venus upper atmosphere ("four-day circulation"), various attempts have been made to explain its generating mechanism. Gierasch (1975) proposed an explanation based on the upward transports of angular momentum by the meridional circulation; if latitudinal distribution of zonal winds is close to a solid body rotation by the predominant action of horizontal mixing, upward transport of angular momentum by the meridional circulation in the equatorial regions predominates over downward transport in the polar regions. As a result, the atmosphere in the upper layer can be accelerated by the net upward transports of angular momentum. On the basis of this idea, Gierasch (1975) calculated a balanced state with fast zonal flow in a diagnostic model. Matsuda (1980, hereafter referred to as M) extended Gierasch's idea to the prognostic model in which the zonal flow is dynamically coupled with the meridional circulation. Using the twolayer model, M examined the characteristics of stationary solutions obtained for a wide range of some external parameters. It has been shown in M that the dynamic coupling between the zonal flow and the meridional circulation can lead to the multiple equilibrium states for some range of external parameters. That is, three states appear as a steady solution; the first solution is characterized by the fast zonal flow and weak meridional circulation, the second one is char-

2 246 Journal of the Meteorological Society of Japan Vol. 60, No. 1 acterized by the strong meridional circulation and the third one has characteristics intermediate between these two states. The first solution corresponds to the four-day circulation in the Venus atmosphere, while the second one is interpreted to express the circulation between the day side and night side (see M). The appearance of the multiple equilibrium states is one of the most interesting results of M. Although M examined various solutions obtained in his model in detail, some important problems have been left unsolved. One of them is concerned with the stability of the above three stationary solutions. By speculation, M predicted that the first and the second solution were stable, and the third solution was unstable. For the application of the results of M to the Venus atmosphere, the determination of the stability of these solutions is very important. Nevertheless, this prediction is not yet verified. The second problem is the crudeness of the two-layer model used in M. We can not calculate the superrotation rate with sufficient accuracies by the two-layer model and hence cannot compare the result with the observation. Lastly, the time evolution of zonal flow has not been investigated at all in M. The time evolution should be examined because the generating mechanism of the fast zonal flow can be most clearly understood by examining its time evolution. Moreover, the examination of evolution of the system becomes more important when multiple equilibrium states exist in the system, because each stable equilibrium states will be attained through different evolutional course. The purpose of the present study is to solve these problems. For this purpose, we employ a five-layer model in place of the two-layer model used in M and obtain stationary solutions by numerical time integration. Except the vertical resolution, the present model is the same as in M. That is, our basic equations are the same truncated spectral equations as used in M. The Boussinesq approximation is also employed. Finite difference scheme in the vertical direction is obtained in the same manner as in M. We integrate this system from a suitable initial state by the use of the Runge-Kutta-Gill method, till a state of the system converges to a stationary state. Results of the calculations will be described in the following three sections. In section 2, a spin-up process of the zonal flow will be investigated by assuming infinite horizontal viscosity. This assumption is not realistic. However, fol the purpose of observing only the spin-up process without any complication associated with the problem of multiple equilibria, it is advantageous to employ this simplification. In section 3, assuming finite horizontal viscosity, calculation will be made for a wide range of some external parameters. By comparing the parameter dependence of multiple equilibrium states obtained in this study with that in M, the prediction in M on the stability of the multiple equilibrium states will be verified. In section 4, each physical process generating multiple equilibrium states will be examined. Section 5 is devoted to the concluding remarks. 2. Spin-up process of the fast zonal flow In this section, we investigate the process of spin-up of the fast zonal flow by upward transports of angular momentum by the meridional circulation. As mentioned in the introduction, we assume that horizontal viscosity is infinite in this section. Except the horizontal viscosity, two fundamental parameters are involved in the present model. One is "Grashof number" Gr, which represents the magnitude of latitudinal differential heating. Gr is written as Gr= (*gq/*3)(h7/r2) where * is thermal expansion coefficient, g is the acceleration of gravity, Q is latitudinal differential heating, * is the vertical eddy diffusion coefficient, R is the radius of the planet and H is the depth of the fluid layer under consideration. The other fundamental parameter is the ratio of the planetary rotation period * to the relaxation time of vertical diffusion *; */ =(2*/*)/(H2/*), where * * is the angular velocity of the planet. The value of * in the Venus atmosphere has not been determined by observations, and the value of H is also somewhat obscure because the upper atmosphere (or cloud layer) in the Venus is not so clearly bounded. Owing to this uncertainty of H it is very difficult to determine Gr, which involves 7-th powers of H. Thus, at least in the present stage, we can not find the correct values of Gr and */* for the real state of upper atmosphere. However, since the purpose of this section is only to examine the process by which fast zonal flow is formed, we use Gr=1.75*107 and */*= 0.2* which are obtained by assuming *= 5*104cm2/s and H =14km. With these values, as will be shown below, we can obtain a zonal flow which has a plausibly large velocity at the equator. Other parameters, which are fixed

3 February 1982 Y. Matsuda 247 throughout the present study, are as follows; ratio of vertical eddy viscosity to vertical eddy thermal diffusivity (eddy Prandtle number) is assumed to be unity, coefficient of Newtonian cooling c is 1K/day and averaged static stability is 0.1K/km. Certainly this value of ** is fairly underestimated compared to the arithmetical mean of * in stable layers observed by Pioneer Venus (Seiff et al. 1979). However, according to this observation, there exist some layers with negative * (i.e. unstable layer) inside the cloud layer. Hence, it is reasonable to adopt rather small value as an effective static stability, because the existence of the unstable layers must strongly reduce the effect of the stable layers. We carry out time integrations of our model with the above parameter values. The initial state is assumed to be a motionless state with uniform temperature. Time evolution of super rotation rate (U/R*) in the uppermost layer and the magnitude of the meridional circulation are depicted in Fig. 1. From this figure, we can understand the spin-up process of the zonal flow as follows. First, the latitudinal differential heating which acts on a motionless state excites the strong meridional circulation. It is seen that the set-up time of the meridional circulation is of the order of 10 days. This is roughly equal to the overturning time of the meridional circulation. In this early stage production of vorticity due to the latitudinal temperature difference in a meridional plane can be balanced only by the viscous diffusion of the meridional cell. After this early stage, magnitude of the zonal flow is gradually increasing. This is caused by upward transport of angular momentum by the meridional circulation which is already built up. It is found that a spin-up time by this process reaches 1000 days, which is much longer than the set-up time of the meridional circulation. This spin-up time is essentially controlled by the relaxation time of vertical diffusion *, because the final state of the process consists of a balance between downward transports of angular momentum by vertical diffusion and upward transports of the meridional circulation. In fact, in this calculation * is about 400 days, which is comparable to calculated spin-up time. Note that the magnitude of the meridional circulation is gradually decreasing with the increase of the zonal winds. This shows that the vorticity balance mentioned above in the early stage is gradually transformed into the balance of another type in which a torque of buoyant forces due to the latitudinal temperature difference is sustained by a moment due to the gradient of centrifugal force of zonal winds with vertical shear. Next, we examine the vertical profiles of zonal winds at the equator. In Fig. 2, the profiles at 100 days and 3900 days are compared. The former profile shows that the atmospheric layers except the uppermost one rotate more slowly than the solid part of the planet. (Note that velocity of zonal winds is measured relative to the reference fixed on the solid planet. Positive sign is directed towards the rotation of the planet.) The meridional circulation suddenly induced in the initial motionless state transports angular momentum from the lower layer to the upper layer. As a result, the lower layer loses some parts of its angular momentum and begins to rotate more slowly than the planet. Then, angular momentum is supplied from the solid part to the lower atmosphere by the action of vertical eddy diffusion. This means that a net increase of angular momentum of atmospheric layers. This momentum supplied to the lower layer is in turn transported upwards by the men- Fig. 1 Time evolution of the zonal flow and the meridional circulation in the uppermost layer. The zonal flow is shown by its velocity at the equator normalized by the velocity of planetary rotation R*. Magnitude of meridional circulation is shown by representative meridional velocity normalized by R* but with a numerical factor. Time scale is measured by an earth day. Fig. 2 Vertical profiles of the zonal velocity at 100 days and 3900 days.

4 248 Journal of the Meteorological Society of Japan Vol. 60, No. 1 dional circulation. Repeating this process, angular momentum is gradually accumulated in the upper atmosphere. The vertical profile of zonal winds at t=3900 days illustrates the final state formed in this way. 3. Nature of multiple equilibrium states In section 2, we have assumed that horizontal viscosity is infinite. For more general discussions, we shall relax this assumption in this section. Thus, in addition to Gr and */*, there appears a new parameter * which is the ratio of relaxation time of horizontal diffusion to that of vertical diffusion, * = (R2/*)/(H2/*). (a) Effect of horizontal viscosity We examine the effect of horizontal viscosity in this subsection before the detailed analysis of the solutions obtained in the present model. Ac cording to M, the maximum value of zonal wind is inversely proportional to *. Thus, sufficiently small * is necessary for a zonal wind velocity to attain about 60 times of the planetary rotation speed (i.e, the observed value). At the same time, if * is small, there can appear multiple equilibria for some range of Gr, and this range of Gr becomes wider as * decreases. However, M can not indicate any definite value of * necessary for a sufficiently fast zonal flow, because of the crudity of the two-layer model. Here, we shall examine quantitatively the effect of *. For this purpose, we calculate equilibrium states for several values of * by time integration of the present five layer model, with fixing */* at 0.2*. The results for the zonal winds in the uppermost layer at the equator are shown as functions of Gr for three values of * in Fig. 3 and Fig. 4(b). When * =10-1 (Fig. 3(a)), the zonal wind Fig. 3 Velocity of the zonal flow of stationary solutions as a function of Gr in the case of *= 10-1(a) and in the case of *=10-3(b). The value of */* is assumed to be 0.2*. Values obtained by numerical computations are shown by X. Fig. 4 Velocity of the zonal flow as a function of Gr in the case of *=10-2. (a) is the fast planetary rotation case (*/*=1/15*, (b) is the standard rotation case (*/*=1/5*) and (c) is the slow rotation case (*/*,=2*). Values obtained by numerical computations are shown by *. A broken line indicates unstable solutions.

5 February 1982 Y. Matsuda 249 speed cannot exceed about 5R* (5 times as large as the planet rotation) even at its maximum value (at Gr*1.7*107). In addition, we have only single steady state for all Gr's. If * =10-2 (Fig. 4(b)), we have three steady state solutions (multiple equiliblia) for some range of Gr, and the largest zonal wind speed on the upper branch reaches about 25R* at Gr=1.9*107. When * =10-3 (Fig. 3(b)), we have again the multiple equilibria for some range of Gr (different from that in case of * =10-2), and the zonal wind on the upper branch can exceed the observed value (60R*). Note that the functional form of the zonal winds velocity (i.e., a twovalued function in some range of Gr and a singlevalued function in other range of Gr) in this case (* =10-3) is analogous to that in * =10-2, in spite of a considerable difference in the value of zonal wind velocity itself. From the above results, we can conclude that =10-2 is sufficient for the appearence * of multiple equilibrium states while * =10-3 is necessary for the appearence of the super-rotation rate as large as the observed one. (b) Dependency of the solutions on Gr and /* * In order to discuss the characteristics of stationary solutions in details, we calculate the solutions as a function of Gr and */*, fixing at By calculating with two values * of /* different from that in the previous section, * we shall treat the following three cases; (a) the fast planetary rotation case (*/* = 0.2*1/3) (b) the standard planetary rotation case (*/*= 0.2* same as that in the previous section) and (c) the slow planetary rotation case (*/*= 0.2*2).* The zonal velocity and meridional velocity of stationary states are shown by solid lines in Figs. 4 and 5, respectively. Note that these figures correspond to Figs. 7 and 8 in M which are schematic illustrations of U= U(Gr, */*) and V = V(Gr, */*). (U: velocity of zonal winds at the equator, V: representative velocity of men- * In connection with the application to the Venus atmosphere, it is preferable to regard */* as a parameter expressing the magnitude of vertical diffusion rather than as a planetary rotation speed, because the rotation speed of Venus is exactly known to us. However, for simplicity, we refer to */* as a parameter representing planetary speed of rotation in the following. Fig. 5 Representative velocity of the meridional flow normalized by R* but with a numerical factor. (a) is the fast planetary rotation case (*/*= 1/15*), (b) is the standard rotation case (*/*=1/5*) and (c) is the slow rotation case (*/*=2*). Values obtained by numerical computations are shown by X. The stationary solution with larger (smaller) magnitude of the meridional circulation in (a) and (b) is identical to that with smaller (larger) velocity of the zonal flow in Fig. 4(a) and (b), respectively. A broken line indicates unstable solutions. dional flow). In the case of slow planetary rotation (Fig. 4(c) and Fig. 5(c)), we see that a multiplicity of equilibrium states does not occur. It is also found that the speed of the zonal wind and the meridional wind as a function of Gr is similar to M's result of the slow rotation case which are shown in Figs. 7(*) and 8(*) in M. That is, the

6 250 Journal of the Meteorological Society of Japan Vol. 60, No. 1 results in M and those in the present calculations have the following three characteristics in common; (i) there exists only a unique solution in the whole range of Gr. (ii) the zonal wind velocity in the upper layer has a maximum value at a certain value of Gr, and (iii) the magnitude of meridional velocity is monotonously increasing function of Gr. i.e. almost proportional to Gr. This result (iii) indicates that the meridional circulation is determined almost directly from the balance between the torque of buoyant force due to the latitudinal temperature difference and that of frictional force acting on the meridional cell. The magnitude of zonal wind is determined by the upward transports of angular momentum by the meridional circulation already established. Thus, this type of dynamic balance is nothing but the direct cell balance defined in M. Next, we examine the solutions obtained in the standard (b) and the fast planetary rotation case (a). (See solid lines in Figs. 4(a, b) and 5(a, b).) It is common to these two cases that one of the two coexisting solutions has a large zonal flow and a weak meridional circulation, while the other has a strong meridional circulation and a weak zonal flow. In the latter type of the equilibrium states, it is seen that the magnitude of the meridional velocity is almost proportional to Gr. As in the slow planetary rotation case, this proportionality indicates that the dynamic balance of the equilibrium states is the direct cell balance. On the other hand, in the former type of the equilibrium states, diffusion of the weak meridional cell is negligible in the balance of vorticity in a meridional plane. Hence, the torque of buoyant force due to the latitudinal temperature difference must be balanced by the vertical gradient of the centrifugal force due to the large zonal flow. This type of dynamic balance was called as thermal wind balance of Venus type in M, (See section 4 for the detailed analysis of the time evolution to the above two types of the equilibrium states.) In Figs. 4 and 5, when the value of Gr becomes large, the solution of thermal wind balance of Venus type disappears and only the solution of direct cell balance remains to exist. Evidently, the effect of the centrifugal force due to very small zonal winds in the solution becomes negligible in the vorticity balance n a meridional plane. On the other hand i, when Gr becomes small, the solution of direct cell. balance disappears and only the solution of thermal wind balance of Venus type remains to exist. (c) Comparison with the results in M. Next, we compare the results described above with those in M. Fig. 6 illustrates schematically the zonal velocity of the stationary solutions in the fast or standard (moderate) planetary rotation case in M. As shown in this figure, in M three stationary solutions coexist in some range of Gr. The first (the largest zonal wind) of the three solutions is of thermal wind balance of Venus type, the second (the smallest zonal wind) is of direct cell balance and the third (a moderate zonal wind) has characteristics intermediate between these two solutions. It is predicted in M that the first and the second solutions are stable and the third one is unstable. * Comparing Fig. 4(a) and (b) with Fig. 6, we see that Gr dependence of the zonal velocity of each solutions in this study is analogous to that in M. That is, the two coexisting equilibrium states realized in the present study (i.e., the state with the fast zonal flow and the state with the strong meridional circulation) correspond to the first and the second solution in M, respectively. It is noted that we have only two equilibrium states in this study, while in M there exist three equilibrium states. This does not mean that the present model excludes the third solution in M. Rather, it should be emphasized that the third solution can not be attained by numerical time integration, because it is unstable as predicted in M. (The unstable solution anticipated for the present model is arbitrarily depicted in Fig. 4 and 5 by broken lines.) Thus, we can conclude that the stability of the solutions predicted in M is correct. Fig. 6 Schematic illustration of the zonal velocity as a multi-valued function of Gr in the result obtained by M. * On the basis of Matsuda's study (1981), this prediction can be directly verified without any calculation. As this technic can be applied to various models involving multiple equilibrium states (for example, the model treated by Charney and DeVore (1979)), we briefly describe this demonstration in the Appendix.

7 February 1982 Y. Matsuda 251 Lastly, we mention the difference in the result between the fast planetary rotation case and the standard planetary rotation case. In Figs. 4 and 5, it is seen that the extent of the range of Gr in which multiple equiliblia appear is larger in the former case than in the latter case. The same may be said of the value of Gr itself. We should notice that these differences in the present model is already seen in Figs. 7 and 8 in M. 4. Time evolution to the multiple equilibrium states In M, it has been mentioned that it depends on the initial conditions whether the fast solution or the second solution can be attained. In this section, we shall examine this problem by assuming two different ways of external heating imposed on the initial motionless state with uniform temperature. In case (a) a latitudinal heating difference corresponding to Gr=1.75*107 is suddenly imposed on the system (Fig. 7(a)), while in case (b) the system is gradually heated to the same heating difference after a continuous increase of Gr over Fig. 7 Time evolution of the zonal flow and the meridional circulation in the standard rotation case with Gr=1.75*107. In (a) heating corresponding to this value of Gr is imposed from the beginning while in (b) the heating is gradually increased to reach this value days (Fig. 7(b)). As shown below, this difference in heating makes a state converge to an entirely different equilibrium state from each other. In case (a) (Fig. 7(a)), a large external heating is suddenly imposed on the motionless state, so that a strong meridional circulation is immediately induced. Note that the set-up time of the meridional circulation is roughly equal to the overturning time of the meridional circulation, namely it is comparatively short. Once a fast meridional circulation corresponding to the given Gr is formed, meridional pressure gradient is sustained almost only by a viscous force acting on this meridional cell. Namely, fast zonal winds are not needed for sustaining the meridional pressure gradient. Moreover it is impossible for zonal winds to grow, because finite horizontal viscosity assumed here can not diffuse back the distribution of the angular momentum to the rigid body rotation against the horizontal transport by such a fast meridional flow, and as a result the accumulation of angular momentum due to its upward transport by the meridional circulation does not occur. Namely, the mechanism proposed by Gierasch (1975) can not work. This state in which a strong meridional circulation predominates is a self-consistent balanced state. Evidently, the type of balance in this state is of direct cell balance. On the other hand, in case (b) (Fig. 7(b)) the external heating is gradually increased. Hence, in the early stage only slow meridional circulation corresponding to small Gr are formed. Then, even a finite horizontal viscosity is able to diffuse the angular momentum transported to the polar regions by the slow meridional flow back towards the equatorial region. In short the mechanism proposed by Gierasch (1975) can yield a zonal flow. Once the zonal flow is formed, the meridional pressure gradient becomes to be balanced by the centrifugal force or. Coriolis force of the zonal flow. Hence, the intensification of the meridional circulation is not necessary in response to the increase of meridional pressure gradient caused by that of Gr. Then, Gierasch's mechanism can remain working. In turn this mechanism serves for a production of fast zonal winds which is needed for the balance with an increased meridional pressure gradient. Finally this state converges to the state with fast zonal flow and slow meridional circulation, as shown in Fig. 7(b). Evidently this state is also a self-consistent solution. The type of the dynamic balance in this

8 252 Journal of the Meteorological Society of Japan Vol. 60, No. 1 state is evidently the thermal wind balance of Venus type. Thus, the processes generating the multiple stationary states turn out to be clear. The stationary solutions with and without fast zonal winds described in section 4 have been obtained in these ways. 5. Concluding remarks In this article, we have examined several problems which were obscure in M. First, the spin-up process of fast zonal flow was examined by time integration of our five layer model with infinite horizontal viscosity. The calculations show that the fast zonal flow is gradually formed over 1000 days by the gradual accumulation of angular momentum in the upper layer. This angular momentum is supplied from the slowly rotating solid planet and is transported upward by the meridional circulation. Next, it is found that the super-rotation rate as large as the observed value (i.e., 60R*) appears in our model when * (a measure of the magnitude of horizontal viscosity) is reduced to Thirdly, comparing the multiple solutions obtained in this study with those in M, we verified the prediction in M on the stability of the three equilibrium states. Lastly, by assuming two different ways of external heating, we have shown the difference in the physical processes to the equilibrium state having fast zonal flow and the one having strong meridional circulation; if the heating is suddenly imposed, the latter state is attained, while a gradual heating leads to the former state. In M, it is suggested that the Venus atmosphere corresponds to the parameter range of the multiple solutions and the four-day circulation can be regarded as one of the multiple solutions. Indeed, the multiple equilibrium states anticipated in M are reproduced in the present model. However, Fig. 4 shows that the range of Gr in which the multiple solutions appear is rather limited. Conversely, fairly large super-rotations can be realized outside the multiple solution range of Gr. Hence, we can not necessarily exclude the possibility that the unique solution with the fast zonal flow corresponds to the four-day circulation and a strong direct cell can not exist in the Venus atmosphere. In order to determine whether the Venus atmosphere corresponds to the parameter range admitting the multiple equilibrium states or not, it is necessary to find a correct value of Gr and */*. As already mentioned, we can not know a correct value of them in the present day. The determination of its value is a subject in future. In the present study as in M, the Boussinesq approximation is assumed. This approximation does not so seriously affect an exact determination of the super-rotation rate. In our mechanism, the super-rotation rate is determined from the balance between upward transports of angular momentum by the vertical motion of the meridional circulation and downward transports of angular momentum by viscous diffusion. This balance can be written as where the conventional notation is used. Integrating this equation, we can drop out * (density) from this balance without using the Boussinesq approximation. In other words, the Boussinesq approximation (i.e., * = const.) gives the same expression of the angular momentum balance as that derived without using the Boussinesq approximation. In spite of this elimination of *, the density variation with pressure (i.e. altitude) have an influence on this balance through * (vertical velocity). That is, for the same vertical distribution of heating, vertical velocity in the upper layer must be much reduced by the Boussinesq approximation. This difference in vertical velocity may contribute to the super-rotation rate. Hence, it is desirable to calculate the super-rotation rate by a more realistic model without the Boussinesq approximation. In this study, we find that * =10-3 is necessary for generating the super-rotation rate as large as the observed one. The large horizontal mixing thus postulated is almost only one assumption which is not yet a sound foundation in our mechanism. Thus, our subsequent subject is to investigate a mechanism yielding such a large horizontal mixing. Acknowledgements The author wishes to express his thanks to Prof. Matsuno for appreciations and discussions. He is also much indebted to the reviewers for valuable comments. He acknowledges Dr. Satomura for his careful reading. The author would like to express his sincere thanks to Prof. Uryu and Dr. Miyahara who kindly give many valuable comments on this paper.

9 February 1982 Y. Matsuda 253 Appendix: Determination of the stability of the multiple equilibrium states based on the general theory of the classification of critical points In this appendix, we examine the stability of three equilibrium states obtained in M from a different viewpoint from section 3(c). Concerning the instability problem in fluid dynamics, Matsuda (1981) has classified the transition between two equilibrium states due to the change in external parameters mainly into two categories; one is the transition accompanying a breakdown of symmetry of a state, the other is the transition in which the symmetry of a state is preserved. An example of the former is Benard type convection; a motionless state which has a translational symmetry in a horizontal plane changes into an asymmetric state of steady cellular convection as Rayleigh number Rn (external parameter) exceeds a critical value Rac. In this type of transition, the symmetric equilibrium state changes its stability at the critical point; if Ra <Rac, the symmetric motionless state is stable, if Ra>Rac 'the asymmetric steady cellular convection and the symmetric state can exist as steady solutions, but only the former is stable. On the other hand, we can regard the present stability problem as an example of the symmetry preserving transition, because we have treated only axially symmetric motions. In this type of transition, it can be shown that two equilibrium states near a critical point are necessarily a pair of a stable and an unstable steady solutions. On the basis of this result on the symmetry preserving transition, we can determine, by an inspection, the stabilities of three steady solutions obtained in M. In Fig. 6, we see that there are two critical points Gr=Gr1 and Gr='Gr2, and that near Gr= Gr2 there are two steady solutions with the largest U/ R Q and the intermediate one and near Gr='Gr1 two steady solutions with the intermediate U/RQ and the smallest one. Thus, the following two cases are possible; (i) the intermediate state is unstable and the other two are stable, or (ii) the former is stable and the latter two are unstable. The choice from the above alternatives can be made in the following way. In our model Gr is the only source which can induce the state of the system from a motionless state, so that the unique solution for very small Gr has only very small velocity. As a result, the action of viscosity predominates over the action of nonlinear terms in this solution. Hence the unique solution for very small Gr must be stable. It is noticed that the unique solution for very small Gr is continuous, without crossing any critical points, to the solution which has the largest U/RQ in the three coexisting solutions. (See Fig. 6.) This means that the stability of the solution with the largest U/RQ is also stable. Thus, adopting the former choice, we can conclude that the intermediate solution is unstable and the other two are stable. References Charney, J. G. and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci., 36, Gierasch, P., 1975: Meridional circulation and the maintenance of the Venus atmospheric rotation. J. Atmos. Sci., 32, Matsuda, Y., 1984: Dynamics of the four-day circulation in the Venus atmosphere. J. Meteor. Sac. Japan, 58, :, Classification of critical points and symmetry breaking in fluid phenomena. (unpublished.) Seiff, A., et al., 1979: Thermal contrast in the atmosphere of Venus: initial appraisal from Pioneer Venus probe data. Science, 205,

10 254 Journal of the Meteorological Society of Japan Vol. 60, No. 1