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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 five-layer 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 (nor-malized by the vertical one) 8 is reduced to 10-2. The super-rotation rate as large as the observed one is realized in some solutions when 8 is reduced to 10-3. 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 circu-lation"), 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 bal-

anced 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 two-layer 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-

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 be-tween these two states. The first solution cor-responds 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 ob-tained 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 super-rotation 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 under-stood by examining its time evolution. More-over, the examination of evolution of the system

becomes more important when multiple equili-brium 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 investi-

gated 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, as-suming finite horizontal viscosity, calculation will be made for a wide range of some external

parameters. By comparing the parameter de-pendence 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 pro-

cess generating multiple equilibrium states will be examined. Section 5 is devoted to the con-cluding 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 trans-

ports 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 some-

what 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


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 arith-metical 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 heat-ing 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 meri-dional 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 circu-lation 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 dif-

fusion and upward transports of the meridional circulation. In fact, in this calculation * is about 400 days, which is comparable to calcu-lated 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 circu-

lation 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.


dional circulation. Repeating this process, angu-lar 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, sufficient-ly 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 * neces-sary 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 plane-

tary 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 .


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 (dif-ferent 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 two-valued function in some range of Gr and a single-valued 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 10-2. 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 com-

putations 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 meri-dional 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


results in M and those in the present calculations have the following three characteristics in com-mon; (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 equili-brium 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 bal-ance 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 tempera-ture 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 ther-mal 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 solu-tion becomes negligible in the vorticity balance

in a meridional plane. On the other hand , when Gr becomes small, the solution of direct cell . balance disappears and only the solution of ther-mal 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 be-tween 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.


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 as-suming two different ways of external heating imposed on the initial motionless state with uni-form temperature.

In case (a) a latitudinal heating difference cor-responding 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.

4000 days (Fig. 7(b)). As shown below, this dif-ference 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 immediate-ly 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 pres-sure gradient. Moreover it is impossible for zonal winds to grow, because finite horizontal viscosity assumed here can not diffuse back the distribu-tion of the angular momentum to the rigid body rotation against the horizontal transport by such

a fast meridional flow, and as a result the ac-cumulation 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 predomi-nates is a self-consistent balanced state. Evident-

ly, 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 meri-dional pressure gradient becomes to be balanced

by the centrifugal force or. Coriolis force of the zonal flow. Hence, the intensification of the meri-dional circulation is not necessary in response to the increase of meridional pressure gradient caused by that of Gr. Then, Gierasch's mecha-nism can remain working. In turn this mecha-nism 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 solu-tion. The type of the dynamic balance in this


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 ob-served value (i.e., 60R*) appears in our model when * (a measure of the magnitude of hori-zontal viscosity) is reduced to 10-3. 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 atmos-

phere 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. How-ever, 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 circula-tion and a strong direct cell can not exist in the Venus atmosphere. In order to determine whether the Venus atmosphere corresponds to the parame-ter 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 meri-dional circulation and downward transports of angular momentum by viscous diffusion. This balance can be written as

where the conventional notation is used. Inte-grating 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 approxi-mation. 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 distribu-tion 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. Sato-mura 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.


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 accompany-ing 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 hori-zontal 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 equi-librium 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 inter-mediate 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 con-tinuous, 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 inter-mediate 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, 1205-1216. Gierasch, P., 1975: Meridional circulation and the

maintenance of the Venus atmospheric rotation. J. Atmos. Sci., 32, 1038-1044.

Matsuda, Y., 1984: Dynamics of the four-day circulation in the Venus atmosphere. J. Meteor. Sac.

Japan, 58, 443-470. , 1981: Classification of critical points and

symmetry breaking in fluid phenomena. (unpub- lished.)

Seiff, A., et al., 1979: Thermal contrast in the atmosphere of Venus: initial appraisal from Pioneer

Venus probe data. Science, 205, 46-49.

金星大気四日循環の力学の数値的研究

松田佳久

東京学芸大学地学教室

金星大気の四日循環の生成機構を二層モデルを用いて考察したMatsuda(1980)の研究で未解決の問題を5層

モデルを時間積分することに依り調べた。

先ず,水平粘性が無限大の場合,他の外部パラメータの適当な組み合せに対して,高速の帯状流が得られた。

その形成過程を検討して見ると,高速の帯状流は1000,日を越える非常に長い時間をかけて形成されることが判


った。即ち,最初に子午面循環が南北加熱差に依り短時間のうちに形成され,それの角運動量上方輸送の効果に

依り,固体惑星から吸い上げられた角運動量が上層に少しずつ蓄積される過程が計算された。

次に,このモデルの定常解を三つの外部パラメーター(水平拡散の緩和時間,南北加熱差,惑星の回転速度)

の色々の値に対して求めた。複数平衡解は水平粘性の緩和時間(鉛直拡散の緩和時間で割った)が10-2の時に,

既に現われる。一方,上層大気の回転が惑星の60倍(観測値)に達する解が得られる為には,その値が10-3程

度に成ることを要する。このモデルで得られた定常解のパラメーター依存性は先の論文で得られた結果に類似し

ている。両モデルの結果の比較に依り,複数定常解の安定性について先の論文で立てた予想が証明された。即ち

中程度及び速い惑星の回転において,適当な南北加熱差の大きさの範囲で出現する三つの平衡解のうち,高速の

帯状流を持った解と強い子午面循環を持った解は安定であり,両者の中間の性質を持った解は不安定であること

が判った。(この証明はMatsuda(未発表)の流体系において現われる臨界点の分類の一般論を用いると,いか

なる計算も要せずに証明される。この証明は付録に付けた。)

最後に,系の異なった加熱の仕方に対して,上の二つの異なった安定平衡解を得た。即ち,ゆっくりした加熱

に依って,高速帯状流の解を,急激な加熱に依って,強い子午面循環の解を得たが,それぞれの解へ到る異なっ

た過程を調べた。

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