A Study of Finite Amplitude Barotropic Instability

We report cases with a series of disturbances tilted upshear along the shear zone during the Mei-Yu season. The scale involved is much smaller than the local Rossby radius of deformation; we hypothesize the relevance of barotropic instability, and have explored the nonlinear evolution of barotropic instability. The small and finite ampli­ tude theories are reviewed; their relevance to the observations are briefly discussed. Eigenvalues of ideal three and four region models are calculated analytically. A new Fourier Chebyshev nondivergent barotropic model is constructed. With the initial value problem approach, our experiments on barotropic instability with a vorticity strip either have hyperbolic tangent or Bickerly jet type of wind profiles. We studied the time evo­ lution of the shear layers in terms of the formation of fundamental eddies and successive pairing or merging of eddies. The mutual intensification of counter-propagating Ross by. waves (vorticity gradient waves) across the vorticity strip, breakdown of the vorticity strip, local concentration of vorticity and vortex mergering processes were simulated. While the barotropic instability is an efficient way to concentrate vorticity in a small region, the I11ter vortex merging can enlarge the vortex sizes but not the intensity of the resultant vortex. We propose that the concentration and the merging of vortices can create a favorable localized environment within the shear zone for the moist baroclinic processes to operate. As far as the intensity, shape and evolution of the individual vortices are concerned, they are very sensitive to initial background noise (1/100 of mean vorticity). In other Words, there is no predictability in the nonlinear evolution. However, the maximum growth rate and the dominant wavelength of vortex can be predicted from the linear analysis. The Bickerly jet possesses higher growth rate than the hyperbolic tangent case on the f plane. We document chaotic behavior of sudden breakup of shear zon� and associated vortex merging after a couple of regular cycles of wave-mean flow interaction with shear zone maintained. In the thirty-day integration, the vortices on the /3 plane become disorganized and scattered while they still remain well organized on the f plane.

Eigenvalues of ideal three and four region models are calculated analytically. A new Fourier Chebyshev nondivergent barotropic model is constructed. With the initial value problem approach, our experiments on barotropic instability with a vorticity strip either have hyperbolic tangent or Bickerly jet type of wind profiles. We studied the time evo lution of the shear layers in terms of the formation of fundamental eddies and successive pairing or merging of eddies. The mutual intensification of counter-propagating Ross by. waves (vorticity gradient waves) across the vorticity strip, breakdown of the vorticity strip, local concentration of vorticity and vortex mergering processes were simulated. While the barotropic instability is an efficient way to concentrate vorticity in a small region, the I11ter vortex merging can enlarge the vortex sizes but not the intensity of the resultant vortex. We propose that the concentration and the merging of vortices can create a favorable localized environment within the shear zone for the moist baroclinic processes to operate.
As far as the intensity, shape and evolution of the individual vortices are concerned, they are very sensitive to initial background noise (1/100 of mean vorticity). In other Words, there is no predictability in the nonlinear evolution. However, the maximum growth rate and the dominant wavelength of vortex can be predicted from the linear analysis. The Bickerly jet possesses higher growth rate than the hyperbolic tangent case on the f plane. We document chaotic behavior of sudden breakup of shear zon� and associated vortex merging after a couple of regular cycles of wave-mean flow interaction with shear zone maintained. In the thirty-day integration, the vortices on the /3 plane become disorganized and scattered while they still remain well organized on the f plane.

INTRODUCTION
Mei-Yu season in subtropical Asia is the transition period between the dry northeast and moist southwest monsoons. There have been many studies on Mei-Yu fronts and their associated phenomena (see Chen 1992 for a review). An interesting feature of the Mei-Yu season is a series of disturbances growing along the shear zones. These shear zones often extend from the surface up to 700 hPa and are associated with a weak temperature gradient over southern China (Chen and Chang,19 80). Because of the weak temperature gradient and smaller scale involved (smaller than the local Rossby radius of deformation), the significant wind change across the shear zone suggests that barotropic instability may be related to the growth of disturbances along the shear line. Figure 1 shows the 850 hPa wind, temperature and the p velocity of 700 hPa in a 12 hour sequence starting at 12Z, May 13, 1993. The strong baroclinicity and wind shear that resemble the classical hyperbolic tangent wind profi le over south China (center of the map) in this case is obvious. The scale of the shear vorticity is on the order of 10-4 s-1. We note that the wind shear is the strongest at 12Z, May 13, and then decreases as time proceeds. Figure 2 shows the IR satellite picture at OOZ, May 14, which corresponds to time in Figure  lb. Figure 3 gives the 3 hour sequence of IR satellite pictures starting at 13Z, May 13.
Figures 2 and 3 indicate that convections were best organized into a series of disturbances around OOZ, May 14, some twelve hour later than the strong wind shear of 12Z on May 13. The wavelength of the disturbance was about 300 km to 400 km. In addition, Figures 2 and 3 reveal that the disturbances around May 14, DOZ are elongated in the northeast-southwest direction, which are tilted upshear as is required by barotropic instability. Even with the presence of baroclinicity and moisture processes, the small-scale nature plus the features described seem to suggest the relevance of barotropic instability.
Another evidence of the relevance of barotropic instability can be seen from the satellite pictures on June 6, 1992 ( Figure 4 ). In the one-hour sequence of the enhanced satellite picture starting at 07Z on June 6, we observed a series of disturbances grow and decay within a four hour period. In addition, we observed a wavy type of cloud structure surrounding the general area of interest. The synoptic condition did not change significantly during this period, thus we only show the 850 hpa observations at 12Z on June 6, 1992 ( Figure 5). If we are to believe the clouds are signatures of waves or a vorticity fi eld, not only do we see a series of disturbances in the center of the shear zone, but also a wavy structures in both the northern and southern ends of the shear zones. From these satellite images, the wavelength of these waves is about 120 km and the disturbances have a diameter of about 30 km. The ridges or troughs of the waves are north-south oriented, tilted upshear as is required by barotropic instability. In addition, these wavy cloud signatures over the edge of the shear zone are consistent with the argument of counter-propagating Rossby waves (vorticity gradient waves along both edges of a vorticity anomaly strip) interpretation of barotropic instability (Hoskins et al. 1985). The two cases reported here suggest the relevance (or importance) of barotropic instability. However, they do not necessarily imply that the growth of series of disturbances along the shear zone is a pure barotropic case. Certainly, moist baroclinic processes may play an important role. In this paper, we will investigate the relevance of barotropic instability in the development of these disturbances. Conventionally, the barotropic instabilty of a shear flow with or without the downstream variation are studied in the linear sense. Namely, the instability is studied by either the eigenvalue-eigenvector approach or by the numerical integration technique (e.g. Kuo, 1949,

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Crum and Stevens, 1990, among many others). In both approaches, the growth rate and most unstable structure are emphasized. In addition to the review given in Crum and Stevens (1990). more references on barotropic instability can be found in the recent work of Sun and Chang (1992). They apply the numerical integration technique to the hyperbolic type of shear with downstream variation, with the application to Mei-Yu fr onto-cyclogenesis in mind. They also emphasized the predictability of linear stability. In this paper, we take the initial value problem approach, with the inviscid nonlinear evolution in our mind. Thus,    our approach is different also from Williams et al. (1984). They studied the nonlinear effects of barotropic instability in a downstream varying easterly jet. To study the long term equilibrium in their model. frictional effect is included in their mean forcings. Similar to Lesieur et al. (1988), we explore the nonlinear time evolution of shear zone in terms of the fonnation of fundamental eddies and successive pairing and merging of eddies. Unlike Lesieur et aL (1988), our calculation is inviscid and includes the Bickerly jet type of shear flow. Moreover, we emphasize the predictability in the nonlinear evolution. Simulations of chaotic breakdown of shear zones are also performed and discussed. A new spectral Fourier Chebyshev nondivergent barotropic model, which conserves mean enstrophy and mean kinetic energy, was built for that purpose.
In section 2 we perform analytical calculations of eigenvalue of ideal three and four region models. In addition. we review the theories (linear and nonlinear) of barotropic insta . bility and their relevance to observations are also briefly discussed. The Fourier Chebyshev spectral barotropic model is formulated in section 3. The nonlinear evolution of shear zones in terms of eddies interaction is given in section 4. Section 5 summarizes the results and gives concluding remarks.

NONDIVERGENT AND DIVERGENT BAROTROPIC INSTABILITY THEORIES
Before we study finite amplitude barotropic instability numerically, we review non divergent, divergent, linear and nonlinear barotropic instability theories. In addition , we perform linear analysis (eigenvalue and eigenvector approach) of three-region (hyperbolic tangent) and four-region (Bickerly jet) models analytically. The sufficient condition for non divergent linear stability, which can be found in mariy dynamic textboo ks, is included for the completeness of the review and comparison for the f i nite amplitude theory.

Sufficient Condition for Nondivergent Linear Stability
We consider the nondivergent vorticity equation linearized about a basic state zonal state u which is a function of y only. The linearized equation takes the form is the wave-activity (Andrews et aL, 1987). While there is a phase variable, whose time and space derivatives are interpreted as the local frequency and wavenumber, the wave-activity governs the wave intensity. It is the conservation of linear and nonlinear wave-activity we shall derive. Using (2.2b) and the condition of nondivergence in the right hand side of (2.5a), we now write the conservation of wave-activity aA a[uA + Hv'2 -u'2)] a [-u'v (2.6 ) Integration of (2.6) over the domain with vanishing activities along the boundary yields ! J J Adxdy = 0. (2.7) Since the integral in (2.7) must be time invariant, (2.5b) shows that if ( is a monotonically increasing function of y, neither _ry2 nor ('2 can grow in an overall sense. Thus, a necessary condition for instability is that d(/dy have both signs. As has discussed Eliassen (1983) that a frequent wording of this condition, namely that !_he vorticity gradient vanishes somewhere within the domaip., 'is inexact; for one thing, d( / dy does not have to be everywhere a continuous function of y.
The above argument has been generalized in several ways. Baroclinic effects can be included in both quasi geostrophic (Charney and Stem 1962) and semigeostrophic (Eliassen 1983, Magnusdottir and Schubert 1990, 1991 frameworks. Moreover, the analysis need not be limited to parallel shear flow (Andrews, 1983) or even to linear dynamics (Amol'd 1965(Amol'd , 1966Drazin and Reid 1981;Mcintyre and Shepherd 1987;Shepherd 1988aShepherd ,b, 1989. We shall consider the nonlinear extension now.

Sufficient Condition for Nondivergent Nonlinear Stability
To generalize the linear arguments of the previous subsection, we. now consider the nonlinear nondivergent barotropic equation is the absolute vorticity and the wind components ( u, v ) satisfy the continuity equation au av -0 ax + 8y -. (2.10) We divid� the fields into a basic state part and a part associated with wave or eddies, e.g., ((x,y,t ) = ((y) + ('(x,y,t), where the primed variables are departures from the basic state, and are not necessary small amplitude. The x invariant basic state flow is assumed to be a steady solution of (2.8). We consider the case in which ((y) is a monotonically increasing function of y, and thereby define the inverse function jj( () such that y( ((y)) = y. Differentiating this last expression yields flt;(y = 1. As the nonlinar generalization of the small amplitude wave-ac. tivity (2.5b), we now follow Mcintyre and Shepherd (1987), Shepherd (1988a), Haynes (1988) and Schubert et al. (1992) to define  which is a nonlinear generalization of (2.5a)., Using the nondivergent condition (2.10), we can write down counterparts of (2.6), the fi nite amplitude wave-activity conservation equation (2.16) By comparing (2.6) and (2.16), several differences are noteworthy. All the primed quantities in (2.6) are small amplitude , whereas the primed quantities in (2.16) may be of finite amplitude. Where the flux ( uA, v A) appears in the finite amplitude relation (2.16), the flux ( uA, 0) appears in the small-amplitude relation (2.6).
To obtain the nonlinear stability condition, we integrate (2.16) over the domain to yield ! J J Adxdy = 0.

Linear Analysis of Three and Four Region Model
To confirm that a zonal flow is unstable and to find the growth rates and modes of breakdown of the flow, an eigenvalue-eigenfunction calculation is useful. Following the classical approach of Rayleigh (1945, pages 392-394). Gill (1982), Haurwitz (1949) and Guinn and Schubert (1993), let us consider the nondivergent barotropic model on an infinite f plane. For a basic state shear defi ned by and the corresponding relative vorticity is where (o and Y o are constant. The u is a f9ur region idealization of the so called "Bickerly jet" ( -(0y0sech2 ( y/y0)). The u and ( of this model are shown in Figure 6a. Now consider the linearized nondivergent barotropic vorticity equation where t/J is the streamfunction for the perturbation winds u' = -81/J / ay and v' = 8'¢ /ax. in which cases the(' = V2'¢. We shall search for the solution of (2.22) with the basic state of (2.20). Assuming the solution has the form t/J(x,y,t ) = L k c W k c ( y )ei k (x-ct) where k is the real zonal wavenumber, c is the phase speed which can be a complex. Dropping the subscript kc of W k c(Y) for simplicity and performing the discrete Fourier transform to Since the basic vorticity ( is piecewise constant, the perturbation vorticity will vanish every where except alone the line y = 0 and y = ±y0• As solutions of (2.22) which are bounded as ! Y I -+ oo, we have or (2.24a) t/J (x , y, t) = L( '11 8e-k Jy+yal + Wm e -k ly l + W n e-k ly -yol ) e ik(x-ct) ' (2.24b) kc where '11 no Wm and '11 8 are complex constants. The solution associated with the constant '11 m has vorticity anomaly concentrated in y = 0 and the corresponding stream function Hung-Ch i Kuo & Ch ung-Ho Horng decays when it is away from y = 0. Similarly, the solution associated with the constants W n (W 8) has vorticity concentrated in y = yo (y = -y0) and the corresponding stream function decay away from y = Yo (y = -yo). To determine W n• Wm and W 8, we integrate (2.23) over a narrow region centered on either y =O or y = ±y0 and let the narrow region approach zero. This yields dw - where µ -ky0 is the dimensionless wave number and u = kc/ ( a the dimensionless growth rate or frequency. If the basic state' vorticity jumps at the middle and southern interface were removed. the last two terms in ( Note that the effect of these interactions decays with increasing wave number and increasing shear layer width according to the exponential tfecay. Regarding (2.26)-(2.28) as a linear homogeneous system with unknowns '1i n• '1i m and '1i a. we require that the determinant of the. coefficients vanish, which yields the cubic equation The dimensionless growth rate u as a function of µ computed from (2.29) is shown in Figure 7. Also plotted in Figure 7 is the dimensionless growth rate for the three region model we shall now discuss.
The basic state shear for the three region model considered is defined by y o SYS Yo; ( oYo and the corresponding relative vorticity is where (0 and y0 are constant. The u is a three r�ion idealization of hyperbolic tangent basic flow (it = -(0y0tanh(y/y0 )). The u and ( of this model are shown in Figure 6b. As have discussed in Guinn and Schubert (1993), and identical to the approach of the four region model, we have the dimensionless growth rate (2.32) Note that the unstable mode occurs in the three region model if 0 < ky0 ::; 0.6392 and in the four region model if 0 $ ky0 $ 1.8291. The peak instability for the three region In the most unstable mode in the three region hyperbolic tangent type of model the time behavior is e0·2012t;;' ot, so thee-folding time is approximately 5 / (0. Since the vorticity is on the order of 10-4 s-1 on the cases reported in this paper, the e-folding time requires only a couple of hours, which is consistent with the satellite pictures in Figure   3. On the other hand, the most unstable mode in the four region Bickerly jet type of model is e 0 ·2470t;;' ot, so the e-folding time is approximately 4 /(0• In summary, the analysis here indicates that "Bickerly jet" possesses a greater range of instability in wave number space than the .. hyperbolic tangent" case, and is more unstable than the "hyperbolic tangent" case.

Sufficient Condition for Divergent Linear Stability
Some of the nondivergent barotropic instability theorems of the previous subsection can be generalized to the divergent barotropic model and to discretely layered (but not continu ously stratified) primitive equation model (thus no quasigeostrophic assumption required) on the ,8-plane and sphere (Ripa 1983(Ripa , 1991. Considering the following shallow water equation (2.36) The equation for P', obtained by forming the vorticity equation from (2.33)-{2.34) and then eliminating the divergence using (2.35), takes the form where V /Vt is the same operator as used in (2.5a). Again, defining the meridional particle  (2. 4 8a, b) To recover the stability results for the nondivergent bar<?_tropic model from the stability results for the shallow water model we consider the limit gh -+ ooL in wh!ch (2.47b) are satisfied for any finite uo. Then, there is no differel!.. ce between the P and (. and a choice of uo such that uo -u < 0 everywhere leads to d( / dy ::; 0 everywhere as sufficient for stability, while the choice of uo such that uo -u > 0 everywhere leads to d( / dy � 0 every"{_here as sufficient for stability. Therefore, a ne�ssary condition for instability is that d(/dy have both signs (Rayleigh's theorem). If d(/dy = 0 at y = fj and choosing u0 = u(y) leads from (2.47a) to (u(Y) .:.._ u(y))d(/dy < 0 everywhere as a necessary condition for instability (Fjortoft's theorem). The Fjortoft theorem implies that the relative background flow and the vorticity gradient are positively correlated. Namely, the vorticity gradient waves are propagating against the relative background flow in such a way that phase-locking is possible. Finally we note that, unlike the nondivergent barotropic model, in application of shallow water stability theory to atmospheric data or to the interpretation of atmospheric models with continuous §tratification there is a considerable freedom in the choice of mean depth, and reasonable h's should probably exceed 100 m. Calculations from Schubert et al. (1992) i� their Australian summer monsoon study, in consistent with (2.47), indicates that a deeper h allows a faster growth rate for barotropic instability. The horizontal wind shear zones often �xtend from surface up to 700 hPa during the Mei-Yu season, which will give a not too small h (second or third vertical internal mode). However, the growth rate will be smaller than the estimation from the nondivergent barotropic model.

THE SPECTRAL NONDIVERGENT BAROTROPIC VORTICITY EQUATION
The starting point for our numerical model is the nondivergent barotropic vorticity equation in a periodic /3 channel. Because we are interested in internal dynamics depending only on the initial basic wind profile, a periodic channel is used instead of an open boundary condition. Since the flow is constrained to be nondivergent, we can write these equations in the vorticity/streamfunction form This is a closed system in ¢ and (, where ¢ and ( are the streamfunction and vorticity and /3 is the gradient of Coriolis parameter f. We shall solve (3.1)-{3.2) on the domain 0 $ x $ L, 0 $ y $ H, with the assumption that all variables are periodic in x and ¢ = 0 on y= O.H. In the following we discuss an accurate spectral method (Fourier-Chebyshev tau method) for solving the system (3.l)-{3.2).
The simulation of barotropic instability places great dem311 ds on spatial discretization schemes used in simulation models. In the present work we have used a spectral scheme in both horizontal directions. In the x direction, Fourier basis functions are used so that the periodicity is built into each basis function. In the y direction, Chebyshev polynomial basis functions are used; the north and south boundary conditions are not satisfi ed by each basis function, but rather by the series as a whole. (3.8) where A��) is the spectral coefficient of 8A/ 8x and fJ�:i 1 > is the spectral coefficient of 8B / 8y. Some of the details in the derivation of (3.7)-(3.8) are given in Kuo and Schubert (1988). The relation between A��) and A mn (the specttal coefficient of A) is (3.9 ) while the relation between B�� ) and Brtin (the spectral coefficient of B) is jJ(0,1) = _±___ mn He n N I: p=n+1 p+ n odd (3.10) Although the spectral evaluation of y derivatives by (3.10) looks at first sight more difficult than the spectral evaluation of x derivatives by (31), we find that this is not the case. Equation (3.10) yields the (backward) recurrence formula A (0,1 ) A (0,1) -4 A Cn-lBm ,n -l -Bm ,n +l -H nBm,n (n = 1, 2 , · ·· , N -1) (3.11) A(Ol ) A(Ol) with the starting values Bm: N + I = Bm; N = 0. For fixed m, the use of (3.11) allows the N values of iJ��) to be computed in O(N) operations. The transform method (Orszag, 1970;Eliasen et al., 1970) is used in computing the spectral coefficients Arnn and Bmn· To eliminate aliasing error in the quadratic nonlinear terms, 3M points in x and 3N /2 points in y are needed in the physical domain.
In the numerical time integration of the above equations, we must solve (3.7) at each time step. For a given m ( -M s m :5 M), .we regard (3.7) as a linear algebraic system in the N + 1 unknowns (fimn (0 :5 n :5 N), with known right hand side (mn· The matrix structure of this linear system is upper triangular except for the last two rows, which come from the boundary conditions. There are many possible ways to solve (3.7), two of which are discussed by Gottlieb and Orszag (1977, page 119-120). Because (3.7) holds for each m separately, the direct method is a reasonable alternative, a situation which does not exist when Chebyshev expansions are used in both directions.
Using simple model equations, Fulton and Schubert (1987) have investigated the relative merits of various time differencing schemes for Chebyshev spectral methods. When the time step is limited by accuracy rather than stability (a § is apparently the case here), fourth-order schemes are more efficient than second-order schemes. Fulton and Schubert have found the fourth-order Runge-Kutta scheme to be the most useful as a general rule, and it has been used here for the time integration of (3.8).

Hyperbolic Tangent Wind Shear Experiment
The initial condition used here is where u0 = 10 ms-1 and y0 = 150 km. Figure 8 gives the mean vorticity and the associated wind field in physical space. The vorticity is on the order of 10-4 s-1 • ·The westerly to the north and easterly to the south resembles a monsoon trough or Mei-Yu front. The wind profile of Eq. (4.1) satisfi ed the necessary condition of barotropic instability discussed in section 2.
To initiate the instability, we have add�d white noise in vorticity according to DEC workstation random number set 79, with magnitude 1/100 of the initial mean vorticity field. Figure 9 gives vorticity in physical space on the f-plane simulation from day 5 to day 11. The breakdown of a vorticity strip into a pair of eddies of different size is clearly seen. According to the linear analysis in section 2, the most unstable mode has .wavelength-about seven or eight times the shear zone; this is consistent with our simulation. Before the formation of eddies, such as in day 7, the counter-propagating Rossby waves (vorticity gradient waves) near strip edge are obvious. In addition, the phase is oriented in the upshear sense (northeast to southwest directions). After the formation of eddies; we see that the vorticity is concen trated in a localized region. The maximum strength in the smaller eddy in day 9 is about 41 % stronger than the original mean background vorticity. The concentration of vorticity  occurs during the process of shear zone breakup. The later vortex merging will only affect the size of eddies but not the strength. Although our model cannot include the moisture physics, the local concentration of vorticity and the later vortex size merging (enlarging) processes certainly will provide a favorable background for the CISK mechanism to operate, as is in the case of typhoon (Schubert and Hack, 1982).
Note here that the formation of eddies in 7 days comes from the fact that 4500 km domain is used in the simulation. If the same 10-4 s-1 order of vorticity field is used in a 450 km domain, which is more relevant to the observational spatial scale cited in section 1, it will take 0.7 day for the eddies to grow. This is consistent with the satellite images of Figures 2-4. Figure IO is the same as Figure 9 except the random· number set 13 is used instead. The counter-propagating vorticity gradient waves as well as the formation of vortex pairs is clearly seen. However, the time evolution is different from Figure 9. We have also performed calculations with random number �et 73 and 59. Figure 11  As revealed in Figure 11, the strength of the pair vortex go through a couple cycles, with period ranging form 4 to 6 days. Similar cycles are observed as in Figure 11. This indicates the importance of the orientation of eddies against the mean fl.ow. Namely, the wave-mean flow interaction is crucial in the vortex evolution within the barotropic context.
Another evidence of the importance of orientation can be found in the f plane experi ment with initial noise specified as 1.0 x 10-6cos[(i + l)27r/24Js-1tanh(y/y0), where i is the grid index in the x direction . . This noise has the structure of wavenumber 2 with wavelength, which resembles the most unstable mcide (but not the structure). Figure 12 gives the time evolµtion of vorticity from day 2 to day 24 with a 2 day interval. On days 4, 12, 20 we observed the upshear tilt of counter-propagating vorticity gradient waves and the follow up eddy growth. On the other hand, on days 8 and 16 we see the downshear tilt and the follow up mean flow growth. The period of the evolution is about 8 days. Figure   13 gives the vorticity field from day 25 to 30. After 3 cycles of wave-mean flow energy exchange without breaking up the shear zone, Figure 13 indicates the sudden shear zone breakup into the regime similar to Figures 9 and 10. Figure 14 gives the time series of the y-averaged spectral coefficients and the energy conversion term -u 1v1 du/ dy. The sudden breakup of the shear zone after three cycles of wave-mean fl ow energy exchange is obvious.
Together with the case of Figure 11, this simulation suggests that the breakup of shear zones by barotropic instability can be unpredictable and chaotic.
To further explore this chaotic behavior, we perform another experiment with two initial elliptical shapes of vortex superimposed on Eq.(4.1). Figure 15 gives the vorticity field in the physical space for the chaotic experiment on f plane from day 0 to day 17 with one day interval. No white noise has been added in this experiment. Figure 16 is the same as Figure 15 except for day 24 to 40 with one day interval. Figures 15 and 16 reveal that the wave-mean flow interaction is with the rotation of the elliptical shape eddies. When the eddies rotate to the orientation that is upshear (downshear), we see the growth (weaken) of  the eddies. Figure 17 gives the time series of the y-averaged spectral coeffi cients. Similar to Figure 14, we observe the chaotic behavior of sudden breakup of she ar zone and associated vonex merging after a very regul ar four cycles of wave-mean flow interaction with shear zone maintained.
We have also performed tests on the f3 plane. The f3 effect is not relevant to the observations cited in section 1. The {3-plane calculations are used as a comparison to the f-plane result. The results, similar to the /-plane calculations in the initial period, are thus not shown here. However, there are differences in the long term behavior. Figure  18 gives the vorticity field in physical space in days 29 and 30 on j and f3 planes with random number set 13. The dominance of wave number 1 on f plane is clearly seen.
However. small region vortices scattered in northwest-southeast directions on the {3-plane result. We have also performed our test with higher resoluvons and different domain sizes, the results are qualitatively similar. However, the detail time evolution is different case by case; again confi nning our hypothesis that we do not have predictability on the position, shape and intensity of the eddies. They can be quite chaotic. On the other hand. the dominant wavelength of disturbances in all cases is well expected from the linear analysis.
Since Figures 12-17 suggest the importance of eddies orientation against mean flow, the role of non-modal growth, optimal excitation and diagnostics of wave activities will be studied further in the future.

Bic k erl y Jet Experiment
The initial condition used here is the easterly jet that satisfies the necessary condition discussed in section 2 where u 0 = 10 ms -1 and Yo = 150 km.

(4.2)
To initiate the instability, we have added white noise to the mean vorticity. Figure 19 gives vorticity in physical space on days 0, 30 and from day 3 to 12. The initial shear zone consists of strips of negative vorticity (to the north) and positive vorticity (to the south). The vorticity strip broke down into three pairs of eddies with positive and negative vorticity during day 5 to day 7. According to the linear analysis the most WlStable eddy has a wavelength about four times the shear zone, which is consistent with our simulation . Before the fonnation of eddies, such as in day 5, the COWlter-propagating Rossby ·waves (vorticity gradient waves) with upshear tilting (northwest to southeast tilt for negative vorticity stripe and northeast to southwest for positive vorticity stripe) are obvious. After the breakup of the shear zone, we see that the local concentration of vorticity was up to 58% stronger in day 7 (0.93 x 10-4 s-1 to the initial backgroWld vorticity 0.59 x 10-4 s-1), and up to 97% from day 8 and day 11 (l.12 x 10-4 s-1). The vorticity concentration is stronger than the hyperbolic tangent case. This agrees well with our linear analysis in section 2. Namely, the growth rate in the Bickerly jet is stronger than in the hyperbolic tangent case. Finally, we note that the vortex merging process in the simulation is different from the hyperbolic tangent case. On day 30, we have two pairs of positive and negative vorticity vortex distributed in space in a seemingly random fashion. Figure 20 is the same as Figure 19 except for the f3 plane experiment. The ,B-plane experiment gives a very similar picture to the f-plane experiment for the first 12 days. However, the long term behavior of the {3-plane experiment contains small region vortices scattered in a seemling random fashion (day 30).' Figure 21 gives the time series (up to day 30) of y-averaged spectral coefficients from wavenumber 1 to 5 for experiments with different white noises on the f plane (left column) and f3 plane (right column). After the initial dominance of wave number 3 or 4, as expected from the linear analysis, the later nonlinear regime is different case by case. No predictability for the instability seems to be an obvious conclusion.   • · · · 1:s00" · · • · · · · • · · 3ooc,· · · · · · · · · · · 4500 .

SUMMARY AND CONCLUDING REMARKS
We present cases that contain a series of disturbances along the shear zone. In one case the wavy structures resemble the upshear tilted counter-propagating Rossby waves (vorticity gradient waves) on the edge of shear zones. In the other case the disturbances are elongated in the upshear sense as is required by barotropic instability. The horizontal shear in one case decreases after the series of disturbances grows. 4Since the scale involved is much smaller than the local Rossby radius of defonnation, we hypothesize the relevance of barotropic . instability.
The nonlinear evolutions of barotropic instability are studied. We review the small and fi nite amplitude theories. Eigenvalues of ideal three an d four region models are explored.
A new Fourier Chebyshev nondivergent barotropic model is constructed. Using the initial value problem approach, we experiment on barotropic instability with a vorticity strip that either has a hyperbolic tangent or a Bickerly jet type of wind profile. We study the time evolution of the shear layers in terms of the formation of fundamental eddies and successive pairing or merging of eddies. The mutual intensification of counter-propagating Rossby waves (vorticity waves) across the strip, breakdown of the vorticity strip, the local concentration of vorticity and vortex merger processes are simulated. The dominant wavelength is well predicted by linear analysis� In addition, we observe upshear tilted vortices with uneven sizes, in agreement with the satellite images. The local concentration of vorticity is stronger in the Bickerly jet case than that in the hyperbolic case. However, the vorticity concentration only occurs during the process of shear zone breakup. The later vortex merging will only affect the size of eddies but not the strength. Barotropic instability can be viewed as an efficient way to get vortex concentrated in a small region with the later vortex merging as a mean to enlarge the vortex sizes. Both processes give favorable �onditions for the moist convection to grow (CISK mechanism). As far as the nonlinear evolution of the vortex intensity, vortex shape and vortex po sitions are concerned, they are very sensitive to the initial background noise and model resolution. In other words, there is no predictability at all. However, the maximum growth rate and the dominant wavelength of the vortex are well predicted by linear analysis. The Bickerly jet seems to possess higher growth rate than the hyperbolic tangent case. Cases of chaotic behavior are observed in our simulations. The chaos contains a couple of wave-mean flow interaction cycles with the shear zone maintained and the sudden break up of the shear zone. In the thirty-day integration, the vortices on the /3 plane are less organized while they remain well organized on the f plane.