## 1. Introduction

This paper considers nonlinear dynamics of an idealized winter polar vortex in the Southern Hemisphere (SH) stratosphere with a barotropic model on a spherical domain. The SH polar vortex is stronger and less disturbed compared to that of the Northern Hemisphere. In other words, the zonal-mean zonal flow is stronger and planetary waves are weaker in the SH due to weaker forcing of the planetary waves in the troposphere. As a result, a major stratospheric sudden warming event had not been observed in the SH before 2002 (e.g., Baldwin et al. 2003). On the other hand, rather periodic variations of the polar vortex, which are characterized by eastward rotation of the elongated polar vortex and change in shape with a period of about a week or so, are often observed in the SH as analyzed in our previous paper (Hio and Yoden 2004, hereafter referred to as HY04). These features in the SH stratosphere are indicative of the weakly nonlinear nature of the polar vortex in the SH.

The wave–wave interaction between the stationary “planetary wave of zonal wavenumber 1” (hereafter denoted as “Wave 1”) and eastward propagating Wave 2 was investigated in HY04 with the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis dataset over 20 years. The stationary Wave 1 is generated in the troposphere mainly by zonally asymmetric lower boundary conditions and has significant interannual variations (Hio and Hirota 2002). On the other hand, barotropic/baroclinic instabilities of the polar night jet play an important role for the generation of the eastward propagating Wave 2 in late winter (e.g., Manney et al. 1988, 1991). The correspondence of negative or nearly zero meridional gradient of the zonal-mean potential vorticity to large-amplitude events of eastward propagating Wave 2 shown by HY04 (their Fig. 15b) suggests in situ instability of the zonal-mean zonal flow as the generation process. A typical case analyzed in HY04 shows that the transient Wave 1 generated by the wave–wave interaction has comparable amplitude to those of the stationary Wave 1 and the traveling Wave 2 and has a node around 60°S where these primary waves have large amplitudes. The transient Wave 1 travels eastward with the same angular frequency as that of the traveling Wave 2. The zonal-mean polar night jet also oscillates with the same frequency such that it has its minimum when the stationary Wave 1 and the transient Wave 1 are in phase at the polar side of the node.

HY04 also discussed similar wave–wave interactions in another case in August and September 2002, before the unprecedented major stratospheric sudden warming event in the SH. Generally such wave–wave interactions in these months are not as clear as those in late winter due to the slower phase speed of traveling Wave 2. However, the situation was exceptional in the midwinter in 2002 because the seasonal march was much earlier than the other years (Hio and Yoden 2005).

In HY04, similar periodic variations of the polar vortex were also obtained in a numerical experiment with a spherical barotropic model in which the stationary Wave 1 was forced by surface topography, while the eastward propagating Wave 2 was generated by the barotropic instability of a forced zonally symmetric polar night jet. The model is identical to that developed by Ishioka and Yoden (1995, hereafter IY95) if the surface topography is removed. IY95 investigated the flow regimes for two barotropically unstable jet profiles, introduced by Hartmann (1983), by changing three parameters that control the intensity, width, and latitudinal position of the polar night jet. They obtained quasiperiodic vacillation solutions and nonperiodic irregular solutions as well as periodic steady-wave solutions in some parameter ranges.

Such single-layer models have been used by some others to study fundamental dynamics of the winter stratosphere polar vortex, such as planetary wave breaking (e.g., Juckes and McIntyre 1987; Yoden and Ishioka 1993; Thuburn and Lagneau 1999), equilibrium states of initially barotropically unstable polar vortices (Prieto and Schubert 2001), and chaotic mixing process and transport barrier (Mizuta and Yoden 2001). Recently, Rong and Waugh (2004) investigated internal variations of the polar vortex in a shallow-water model with planetary-scale wave forcing by surface topography and relaxation to a prescribed zonally symmetric equilibrium state. Only weak disturbances on the edge of the polar vortex were obtained for small topographic height, whereas full cycles of the breakdown and recovery of the polar vortex were obtained for large topographic height.

Following the two papers of IY95 and HY04, this work aims to investigate the flow regimes in the same model as HY04 for a wide range of the external parameters that give the amplitude of surface topography and the width of the polar night jet. The former controls the magnitude of a forced stationary wave, while the latter influences the magnitude of barotropic instability. We perform over 150 runs of the numerical time integrations for the parameter sweep to show the predominance of the periodic solutions due to wave–wave interactions. The dynamical situation of the polar vortex is close to that of Rong and Waugh (2004) for small topographic height without breakdown of the polar vortex. In such a situation, the eastward propagating Wave 2 generated by the barotropic instability, as well as the topographically forced Wave 1, plays a crucial role through the wave–wave interactions.

The steady-wave solution in the case without topography is a simple periodic solution in which Wave 2 propagates eastward with a constant phase speed, while the periodic solution in the case with topography is complicated by the wave–wave interactions because all the components must synchronize with the same frequency. To understand the complicated periodic solution and vacillation further, we investigate transitions between these solutions thoroughly because time variations are generally simpler around transition points due to the small amplitude of perturbations. A low-order “empirical mode expansion” of the two-dimensional vorticity equation is introduced to diagnose the wave–wave interactions in the parameter range around transitions of the flow regime.

The model and experimental procedure are described in section 2. Dependence of the flow regime on external parameters is shown in section 3. Some examples of transition of the flow regime, for example, from a periodic solution to a quasiperiodic vacillation, are examined in section 4. A discussion is in section 5 and concluding remarks are in section 6.

## 2. Model and experimental procedure

*q*=

*ζ*+

*f*+

*(fh*/

*H*) is the PV,

*ζ*= ∇

^{2}

*ψ*the relative vorticity,

*ψ*(

*λ*,

*ϕ*,

*t*) the streamfunction,

*λ*the longitude,

*ϕ*the latitude,

*t*the time,

*a*the radius of the earth,

*f*= 2Ω sin

*ϕ*the Coriolis parameter, Ω the angular speed of the earth’s rotation,

*h*the height of the surface topography, and

*H*the mean depth of the fluid layer. The right-hand side of Eq. (1) gives nonconservative terms, and

_{f}is the prescribed vorticity for forcing of the zonal-mean zonal flow. The relaxation time

*α*

^{−}^{1}of the Newtonian-type forcing is set to 10 days and an artificial viscosity coefficient

*ν*is fixed at a small constant (=6.43 × 10

^{4}m

^{2}s

^{−}^{1}) to give the dissipation. The material derivative operator

*D*/

*Dt*and the horizontal Laplacian operator ∇

^{2}are given byandrespectively, where the horizontal velocity

**u**is

_{f}= −(1/

*a*) (∂

u

_{f}/∂

*ϕ*), is set to satisfy the necessary condition of barotropic instability (Hartmann 1983):where

*U*is a measure of the intensity of the prescribed zonally symmetric jet,

*B*its width, and

*ϕ*

_{0}its latitudinal position. All of the conditions are the same as in IY95 for the case of

*h*= 0, and the regime diagram for this case in the three-dimensional parameter space (

*U*,

*B*,

*ϕ*

_{0}) is given in IY95 (their Fig. 1). In this work we sweep the parameter

*B*for 3° ≤

*B*≤ 18° with fixed values of

*U*= 240 m s

^{−}^{1}and

*ϕ*

_{0}= 55°. An eastward propagating Wave 2 is generated when

*B*is so small, or the prescribed jet is so narrow, as to satisfy the condition of barotropic instability. Thus, sweeping the parameter

*B*means the change of the features of the eastward propagating Wave 2.

*r*=

*h*

_{0}/

*H*for 0 ≤

*r*≤ 0.1 to investigate dynamical features depending on the forced stationary Wave 1.

We use a spectral model with triangular truncation T85 in spherical harmonics and the fourth-order Runge–Kutta method for time integrations, as in IY95, Mizuta and Yoden (2001), and HY04.

## 3. Flow regimes

### a. Dependence on B and r

Table 1 is a regime diagram that shows the dependence of flow regime on the width of the prescribed zonally symmetric jet for 3° ≤ *B* ≤ 18° and the ratio of the topographic height to the mean fluid depth for 0 ≤ *r* ≤ 0.1.

*r*= 0 without topography, first row in Table 1, we obtain a time-independent solution with no wave (as denoted by

*N*) for wide jets of

*B*≥ 12°. The prescribed jet becomes barotropically unstable for

*B*< 12°, and we obtain a steady-wave solution (

*S*) for 6° ≤

*B*≤ 11°. In the steady-wave solution, Wave 2 travels eastward with a constant phase speed without changing its horizontal structure, and the streamfunction for this component is expressed aswhere a subscript number indicates the zonal wavenumber

*s*. This is a periodic solution with an angular frequency

*ω*.

_{P}*B*= 8° and 6°, while a vacillation solution was obtained for

*B*= 4° in which the elongated polar vortex with Wave-2 component rotates eastward by changing its shape periodically (see Fig. 4 in IY95 for the time variation of the PV field for

*r*= 0). The vacillation solution (

*V*

_{0}) is quasiperiodic with two incommensurable frequencies of the rotation,

*ω*, and shape change,

_{P}*ω*:Table 1 shows that the transition from steady-wave solution (

_{A}*S*) to vacillation (

*V*

_{0}) takes place between

*B*= 6° and 5.5°. Figure 1 shows the wave decomposition of the PV field of a vacillation solution for

*B*= 4°. Time-averaged structure of the traveling Wave 2,

*C*

_{2}(

*ϕ*)

*e*

^{i}^{2}

*, which has an angular frequency of*

^{λ}*ω*= 2

_{P}*π*/10.8 day

^{−}^{1}, is displayed in the bottom right panel. The amplitude modulation of the Wave-2 field,

*A*

_{2}(

*ϕ*)

*e*

^{iωAt}

*e*

^{i(2λ−ωPt)}, is shown for 10.8 days with a time interval of 1.8 days. The pattern at

*t*= 0 day and that at

*t*= 1.8 day are about the same as those at

*t*= 9.0 day and

*t*= 10.8 day, respectively, except for the longitudinal phase. Spectral analysis shows that the frequency of the amplitude modulation is

*ω*= 2

_{A}*π*/9.2 day

^{−}^{1}(see Fig. 6).

*r*≠ 0), four kinds of solutions are obtained, as shown in Table 1: stationary-wave solution (Sta), periodic solution (

*P*), vacillation solution (

*V*), and irregular solution (

*I*). For wide jets of

*B*≥ 9°, a stationary-wave solution (Sta) is obtained, which is a time-independent solution with forced stationary Wave 1:

*B*is reduced, transition from the stationary-wave solution (Sta) to periodic solution (

*P*) takes place for 8° <

*B*< 9°. The value of

*B*for the first transition associated with the barotropic instability of the zonal jet influenced by stationary Wave 1 is reduced largely by the existence of the surface topography, although the value for the transition is not dependent on

*r*for 0.02 ≤

*r*≤ 0.1. As described in detail by HY04 in their section 3, periodic variation of the polar vortex occurs as a result of the interaction between the topographically forced Wave 1 and the traveling Wave 2 due to barotropic instability of the prescribed jet. The streamfunctions for Waves 1 and 2 in the periodic solution are expressed asandrespectively. Note that these traveling waves have the same angular frequency as that of the amplitude modulation of these waves,

*ω*. The zonal-mean zonal flow also oscillates with the same frequency due to periodic variation of the wave driving by the interference between the stationary Wave 1 and the traveling Wave 1. Figures 2a and 2c show the time-mean structures of traveling Wave 2,

_{B}*C*

_{2}(

*ϕ*)

*e*

^{i}^{2}

*, and traveling Wave 1,*

^{λ}*C*

_{1}(

*ϕ*)

*e*, respectively, while Fig. 2b shows the structure of stationary Wave 1,

^{iλ}*S*

_{1}(

*ϕ*)

*e*. The traveling Wave 2 and the stationary Wave 1 are mostly confined between 50° and 70°S with maximum amplitude around 63°S, while the traveling Wave 1 has the maximum around 67°S and a node around 60°S with the secondary maximum around 55°S.

^{iλ}Vacillation solution (*V*) is quasiperiodic with two incommensurable frequencies, as in the case without topography (*V*_{0}), but the time variations of each component are complicated by the interactions with forced stationary Wave 1, as analyzed in the rest of this paper. Transition from periodic solution (*P*) to vacillation (*V*) takes place between *B* = 3.5° and 4° for *r* = 0.02. There is no vacillation solution between *B* = 4° and 5.5° in the case *r* ≠ 0, in Table 1, at an interval of Δ*r* = 0.02, but it does exist for this range of *B*. One can obtain a vacillation solution for finite values of *r* < 0.02. Nonperiodic irregular solutions (*I*) are obtained for *r* = 0.08 and 0.1 with small *B*. The interval Δ*r*, in Table 1, is too large for us to discuss the transition from a periodic solution to an irregular one. Similar spatial structures of Waves 1 and 2 as those shown in Fig. 2 for a periodic solution are still obtained in the irregular solutions. The similarity of the wave structures is indicative that the wave–wave interactions are common features for a wide range of *r* ≠ 0 and *B* ≤ 8° in Table 1.

### b. Regime transitions at B = 4°

We now examine the regime transitions in more detail at *B* = 4°, denoted with bold characters in Table 1, by changing *r* with smaller intervals (Δ*r* = 0.001 for 0 ≤ *r* ≤ 0.042 and Δ*r* = 0.002 for 0.044 ≤ *r* ≤ 0.09). As shown in the bottom part of Fig. 3a, we have vacillation solutions (*V*) for 0 ≤ *r* ≤ 0.015, periodic solutions (*P*) for 0.016 ≤ *r* ≤ 0.078, and irregular solutions (*I*) for 0.08 ≤ *r*. Figures 3a–c show the dependence on the parameter *r* of some quantities associated with (a) the zonal-mean zonal flow, (b) Wave 2, and (c) Wave 1 at *ϕ* = 62.3°S. A dot denotes the time-averaged value of a quantity and the accompanying vertical bar denotes its variation range.

As shown in Figs. 3a and 3b, the time averages of the zonal-mean zonal flow and the amplitude of traveling Wave 2 decrease a little as *r* increases in the vacillation regime. Their variation ranges also gradually decrease with *r*. As shown in the enlargement in Fig. 3c, the time-averaged amplitude of traveling Wave 1 increases almost linearly with *r*, while its variation range has a maximum around *r* = 0.01. The amplitude of stationary Wave 1 is zero when *r* = 0 and increases almost linearly with *r*, while that of the stationary Wave 2 is quite small in the vacillation regime, as shown by the solid lines in Figs. 3c and 3b, respectively.

All variation ranges of the mean zonal flow and traveling Waves 1 and 2 become smallest around *r* = 0.015 where the transition between vacillation solution and periodic solution takes place. The variation ranges of the mean zonal flow and traveling Wave 1 increase gradually and smoothly in the periodic solution regime. They become larger and more sensitively dependent on *r* after the transition to irregular solution at *r* ≃ 0.08 because of the chaotic nature of irregular solutions. On the other hand, the variation range of traveling Wave 2 increases for 0.015 ≤ *r* ≤ 0.052 and then decreases to have a local minimum at *r* = 0.068. It increases again and has large values after the transition to irregular solution.

The time mean of the mean zonal flow decreases gradually as *r* increases and that of traveling Wave 2 decreases over one half in the range of *r* shown in Fig. 3. On the other hand, the time mean of traveling Wave 1 increases in this range. The stationary Wave 2 attains significant amplitude in the periodic solution regime and has a maximum at *r* = 0.052.

### c. Synchronization in periodic solution

Figure 4 shows some examples of the time variations of the zonal-mean zonal flow and Waves 1 and 2 in each regime for eight values of *r* from 0 to 0.09 indicated by downward arrows a–h in Fig. 3a. The top row shows the time variation of the zonal-mean zonal flow at 65.1°S, while the middle and bottom rows show the polar diagrams of the complex amplitude of Waves 2 and 1, such as Figs. 8 and 11 in HY04. The trajectory of each wave component of the relative vorticity [*ζ _{s}*(

*ϕ*,

_{s}*t*)

*e*,

^{isλ}*s*= 1, 2] in a Re

*ζ*− Im

_{s}*ζ*plane gives the time variation of its amplitude and phase by |

_{s}*ζ*

_{s}| and arg

*ζ*, respectively. In Fig. 4,

_{s}*ϕ*is chosen as the latitude at which the traveling wave component has the maximum amplitude, and the value of

_{s}*ϕ*is shown in each panel. To know the phase relationship of these variations, an open circle (○) or plus sign (+) is put at each timing when the zonal-mean zonal flow has a local maximum or minimum value.

_{s}In the vacillation solution without topography, the zonal-mean zonal flow varies purely periodically and the polar diagram of Wave 2 shows a torus structure as shown in Fig. 4a. The torus structure is unique for vacillation and it reflects the modulation of wave amplitude without preference or dependence on the wave phase. Note that the open circles are on the inner side of the torus, while the plus signs are on the outer side. This relationship means a small (large) amplitude of Wave 2 for the maximum (minimum) value of the mean zonal flow. The polar diagram of Wave 1 has no amplitude for *r* = 0, while it shows coexistence of the stationary and traveling components for *r* ≠ 0. The stationary component is denoted by a small dot in each panel, while the traveling one is recognized as the vector difference between a point on the trajectory and the stationary component. Thickness of the torus for Wave 2 gets thinner with *r*, corresponding to the decrease of the variation range of traveling Wave 2 as shown in Fig. 3b.

It is very difficult to distinguish deformation of the vacillation solution due to the existence of surface topography in Figs. 4b and 4c except for the polar diagram of Wave 1, but the deformation becomes clear for larger *r* just before the transition to the periodic solution as shown in Figs. 5a–c for *r* = 0.014 50, 0.015 00, and 0.015 02. A periodic solution shown in Fig. 5d is obtained for *r* = 0.015 04 after the transition. The transition point exists between *r* = 0.015 02 and 0.015 04. Beat of the fluctuation of the zonal-mean zonal flow is seen for vacillation solutions; the modulation becomes small in Fig. 5c and is not found in Fig. 5d. The timings when the mean zonal flow has its local maximum (open circle) or minimum (plus sign) are not very relevant to the phases of Waves 2 and 1 at *r* = 0.014 50 (Fig. 5a), while the timings have preferred phase both in Waves 2 and 1 for *r* = 0.015 00 (Fig. 5b) and 0.015 02 (Fig. 5c). The variation range of the phases denoted by an open circle or a plus sign becomes small as *r* increases to the transition point and the timing is completely locked for the periodic solution (Fig. 5d). In other words, fluctuations of the zonal-mean zonal flow and topographically forced Wave 1 are synchronized with periodic progression of Wave 2 in the periodic solution. These features of the periodic solution remain in the vacillation solutions in the vicinity of the transition point as the preferred phase relationship. Variations of the mean zonal flow and amplitudes of Waves 1 and 2 become rather independent from the progression of Wave 2 in the vacillation solutions as *r* is further reduced.

Around the transition point from vacillation to periodic solution (*r* = *r _{b}* ≃ 0.015 03), the orbits of Waves 2 and 1 in the polar diagrams become nearly circular, consistent with the minima of the variation ranges of the Waves 2 and 1 as shown in Fig. 3. In Fig. 4d for

*r*= 0.016, a perfect circle is included to show the smallness of the deviation of the periodic orbit from the steady wave with a constant amplitude. As

*r*increases, the deviation of the orbits from the circle becomes discernible in Figs. 4f and 4g. The timings when the mean zonal flow has its local maximum or minimum delay in the phase variations of Wavea 2 and 1, as indicated by the counterclockwise shifts of the open circles or plus signs from Figs. 4d to 4g. The phase relationships between the mean zonal flow and traveling waves for large

*r*, as in Fig. 4g, are similar to those shown in HY04 for

*B*= 6° and

*r*= 0.08: Traveling Wave 1 is almost in phase in longitude with the stationary Wave 1 and traveling Wave 2 at the time when the mean zonal flow has its minimum value.

Even in the irregular solution (Fig. 4h) for *r* = 0.09, the mean zonal flow fluctuates rather periodically, and the trajectories of Waves 2 and 1 in the polar diagrams show systematic variations associated with eastward propagation of these waves. The clustering of the symbols indicates that the synchronized phase relationships in the periodic solutions just before the transition remain in this irregular solution.

### d. Power spectra

Power spectral analysis is done to examine the periodic, quasiperiodic, or nonperiodic nature quantitatively in the three regimes obtained in section 3b.

Figures 6a–h show three power spectral densities for the following time series of six solutions given in Fig. 4: (top) the zonal-mean relative vorticity, *ζ*_{0}(*ϕ*_{0} = 62.3°S, *t*); (middle) the amplitude of Wave-2 vorticity, |*ζ*_{2}(*ϕ*_{2} = 62.3°S, *t*)|; and (bottom) the Wave-2 vorticity component at a given point (*λ*_{2}, *ϕ*_{2}) = (0°, 62.3°S), which contains the wave phase information, Re[*ζ*_{2}(*ϕ*_{2}, *t*)*e*^{i2λ2}].

For the vacillation solution (Fig. 6a) without surface topography (*V*_{0} with *r* = 0), both time series of the zonal-mean component and the amplitude of Wave 2 have spectral peaks at the frequency of *f _{A}* = 2

*πω*= 0.112 day

_{A}

^{−}^{1}and its higher harmonics. On the other hand, the power spectral density for the Wave-2 component at a given point has the largest peak at

*f*= 2

_{B}*πω*= 0.0927 day

_{B}

^{−}^{1}and other peaks at the linear combination frequencies of

*f*and

_{A}*f*. These frequencies of the amplitude modulation and the phase propagation of Wave 2 are incommensurable so that the polar diagram in Fig. 4a has a torus structure; this is a fundamental property of quasiperiodic solutions.

_{B}The vacillation solutions in the presence of the surface topography (*V*) have spectral peaks at *f _{A}*,

*f*, and their linear combinations in all of the three power spectra, as shown in Figs. 6b and 6c. The frequency

_{B}*f*corresponding to wave propagation is dominant in the time series at a given point, while the other frequency

_{B}*f*is dominant in the spectra of the zonal mean and Wave-2 amplitude for

_{A}*r*= 0.008 (Fig. 6b). The power at the wave propagation frequency

*f*dominates even in the time series of the zonal mean for

_{B}*r*= 0.014 (Fig. 6c).

For the periodic solutions (*P*), the flow field changes purely periodically at the frequency *f _{B}* of the phase propagation of Wave 2, and thus all the spectra have peaks at the frequencies of

*f*and its higher harmonics, as shown for

_{B}*r*= 0.016 (Fig. 6d) and

*r*= 0.070 (Fig. 6g). Irregular solutions (

*I*) have continuous power spectra with increased power at all of the frequencies, as shown in Fig. 6h for

*r*= 0.090. However, spectral peaks still exist around the frequency

*f*(∼0.11 day

_{B}

^{−}^{1}) of the periodic solutions, indicating that the fundamental feature of the periodic solutions remains, to some degree, in the irregular solution.

To see the variations of the dominant spectral peaks with *r* more quantitatively, we show frequency of power spectral peaks in Fig. 7 (top panel) and corresponding power spectral density (bottom panel) for only three components of *f _{A}*,

*f*, and

_{B}*f*−

_{A}*f*, which are obtained from the time series of (a) the zonal-mean component, (b) the amplitude of Wave 2, and (c) the amplitude of Wave 1, of the relative vorticity at

_{B}*ϕ*= 62.3°S. The frequencies of the dominant peaks of the power spectra do not change very much in the range 0 ≤

*r*≤ 0.02. The vacillation solution (

*V*

_{0}) for

*r*= 0 has a spectral peak only at

*f*, and periodic solutions for

_{A}*r*>

*r*≃ 0.01503 have a spectral peak only at

_{b}*f*. On the other hand, another type of vacillation (

_{B}*V*) in the presence of the surface topography (

*r*≠ 0) has the spectral peaks at the three frequencies,

*f*,

_{A}*f*, and

_{B}*f*−

_{A}*f*.

_{B}Variations of the power spectral densities for these frequencies with *r* are very similar between the zonal-mean component (Fig. 7a) and the amplitude of Wave 2 (Fig. 7b). As *r* increases from zero, the power spectral density for *f _{A}* decreases gradually, while that for

*f*increases much sharply from zero. These two components become comparable in magnitude around

_{B}*r*∼ 0.0145, and then the power for

*f*decreases very sharply to zero as

_{A}*r*approaches the transition point, or the bifurcation point of quasiperiodic solution from periodic solution, at

*r*=

*r*≃ 0.015 03. The power for

_{b}*f*increases continuously as

_{B}*r*increases beyond

*r*. The power spectral densities for these frequencies show a different dependence on

_{b}*r*for the time series of the amplitude of Wave 1 (Fig. 7c).

## 4. Diagnosis on regime transitions

As shown in Figs. 6 and 7, some qualitative changes in the time variations of the zonal-mean component and amplitudes of Waves 1 and 2 take place around *r* = 0 and *r _{b}*. The zonal symmetry of the external conditions breaks when the surface topography is introduced, no matter how small

*r*. This symmetry breaking of the external forcing brings a qualitative change in the vacillation solutions from

*V*

_{0}to

*V*. On the other hand, there is no symmetry breaking of the external forcing around

*r*, but the transition from periodic solution (

_{b}*P*) to quasiperiodic vacillation (

*V*) takes place. In this section, these regime transitions are diagnosed from the viewpoint of wave–wave interactions based on a low-order “empirical mode expansion” of the vorticity equation to understand the fundamental properties of time variations of these solutions.

### a. A low-order empirical mode expansion

*X*(

*λ*,

*ϕ*,

*t*) = Σ

^{2}

_{s=0}

*X*

_{s}(

*ϕ*,

*t*)

*e*

^{isλ}and substitute them into the PV equation (1), we finally obtain the zonal-mean, Wave-1, and Wave-2 components of that equation as follows:andafter neglecting the source and sink terms on the right-hand side of Eq. (1).

*ϕ*= 60°S, as shown in Fig. 2 for example, we can introduce a new meridional coordinate

_{c}*ϕ*′ to represent those with a couple of empirical modes as follows:andwhere Δ

*ϕ*(≃30°) is a meridional extent of the variations,

*s*a zonal wavenumber, and

*m*a meridional mode number.

There is no mathematical rigorousness of this low-order expansion, but we can discuss triad wave interaction in the weakly nonlinear dispersion theory reviewed in HY04 more easily with a simple double Fourier expansion. Note that the expansion in spherical harmonics is not effective in the present situation because the variations are confined in the middle and high latitudes of a hemisphere.

For the interactions in such a low-order system obtained by a double Fourier expansion, we can rely on a classic theory by Lorenz (1963) on the mechanics of vacillation in a rotating annulus experiment with radial differential heating and the studies on the bifurcation properties of a barotropic or two-layer baroclinic flow over topography by the senior author of this paper (Yoden 1983a, b; 1985a, 1985b).

### b. Topographic effect on vacillation around r = 0

*V*

_{0}) without topography at

*r*= 0 is described in the low-order empirical mode expansion asandThe components with coefficients

*C*

^{1}

_{0}and

*C*

^{1}

_{2}constitute a steady-wave solution with a constant Wave-2 propagation with angular frequency

*ω*, while those with

_{B}*A*

^{2}

_{0}and

*A*

^{1}

_{2}represent the periodic variation of the flow field with another frequency

*ω*. In this notation, energy exchange between the mean zonal flow (

_{A}*A*

^{2}

_{0}) and Wave 2 (

*A*

^{1}

_{2}) with the angular frequency

*ω*takes place through the triad wave–mean flow interaction in association with the component of steady Wave 2 (

_{A}*C*

^{1}

_{2}). These components satisfy a three wave resonance relationship: (0, 2) ± (2, 1) = (2, 1), where (

*s*,

*m*) is the wavenumber vector. A schematic diagram of possible interactions between these components can be found in Fig. 1 in Yoden (1983b).

*r*and the linear growth of forced stationary Wave 1 with

*r*from Fig. 3, that is,andfor small

*r*, we can diagnose that triad interactions produce the modulation of vacillation due to the surface topography and we can estimate the dependence of the modulation on

*r*.

- (i) The wave–wave interaction between the stationary Wave 1 (
*S*^{1}_{1}) and the steady eastward traveling Wave 2 (*C*^{1}_{2}) with the angular frequency*ω*produces the eastward traveling Wave 1 (_{B}*B*^{2}_{1}) with the same frequency*ω*. As the amplitude |_{B}*B*^{2}_{1}| is proportional to |*S*^{1}_{1}||*C*^{1}_{2}| from the right-hand side of Eq. (13), it is proportional to*r*from Eqs. (20) and (21). This relationship is schematically written aswhere the parentheses for each component denote the dependence of the amplitude on*r*. - (ii) The interference between the stationary Wave 1 (
*S*^{1}_{1}) and the generated traveling Wave 1 (*B*^{2}_{1}) produces the fluctuation of the zonal-mean component (*B*^{1}_{0}) with the angular frequency*ω*. The amplitude |_{B}*B*^{1}_{0}| is proportional to |*S*^{1}_{1}||*B*^{2}_{1}| from the right-hand side of Eq. (12). Thus, the term*B*^{1}_{0}increases as*r*^{2}: - (iii) The wave–mean flow interaction between the steady eastward traveling Wave 2 (
*C*^{1}_{2}) and the generated zonal-mean fluctuation (*B*^{1}_{0}) produces the second mode of eastward propagating Wave 2 (*B*^{2}_{2}) with amplitude modulation with the angular frequency*ω*. The amplitude |_{B}*B*^{2}_{2}| is proportional to |*C*^{1}_{2}||*B*^{1}_{0}| from the left-hand side of Eq. (14) and increases as*r*^{2}:

*r*:andTo see the dependence on

*r*as stated above directly, the power spectral densities for the frequencies

*f*(plus sign) and

_{A}*f*(closed circle) shown in Fig. 7 are replotted in Fig. 8 on a double logarithmic chart. Both the power of the zonal-mean component (Fig. 8a) and that of the amplitude of Wave 2 (Fig. 8b) for the angular frequency

_{B}*ω*(closed circle) increase with

_{B}*r*along the line

*r*

^{4}. On the other hand, the power of the amplitude of Wave 1 (Fig. 8c) for the angular frequency

*ω*increases with

_{B}*r*along the line

*r*

^{2}. These power laws are consistent with the above arguments, indicating that both

*B*

^{1}

_{0}and

*B*

^{2}

_{2}are proportional to

*r*

^{2}while

*B*

^{2}

_{1}is proportional to

*r*.

### c. Bifurcation of vacillation solution at r = r_{b}

*P*) to vacillation (

*V*) takes place at

*r*≃ 0.01503 as

_{b}*r*is reduced from above as shown in Fig. 5. Because the external forcing is constant with time in this experiment, this could be a bifurcation of a quasiperiodic solution with two incommensurable angular frequencies

*ω*and

_{A}*ω*from a periodic solution with

_{B}*ω*. We diagnose this transition by the low-order empirical mode expansion, describing a periodic solution above

_{B}*r*asandFor

_{b}*r*>

*r*, these components fluctuate at the angular frequency

_{b}*ω*through the wave–wave and wave–mean flow interactions, as described in the previous subsection.

_{B}*r*. When a Hopf bifurcation takes place, generally a small amplitude limit cycle bifurcates from a fixed point of a dynamical system when the fixed point loses stability with respect to a time-periodic perturbation. Near the bifurcation point, the amplitude of the limit cycle increases in proportion to the square root of the increment of the bifurcation parameter. In the present case, the periodic solution with a frequency

_{b}*ω*becomes a fixed point after making a Poincare map for the periodic orbit so that periodic modulation with another frequency

_{B}*ω*appears as a result of the Hopf bifurcation. Assuming an increase of the amplitude modulation of eastward traveling Wave 2 by the square root law asnear the bifurcation point

_{A}*r*with

_{b}*r*<

*r*, we can diagnose triad interactions associated with this amplitude modulation and estimate the dependence of the modulation on (

_{b}*r*−

_{b}*r*).

- (i) The interaction between the steady eastward traveling Wave 2 (
*C*^{1}_{2}) and the amplitude modulation of Wave 2 (*A*^{1}_{2}) produces the fluctuation of the zonal mean component (*A*^{2}_{0}) with the angular frequency*ω*. The amplitude |_{A}*A*^{2}_{0}| is proportional to |*C*^{1}_{2}||*A*^{1}_{2}| from the right-hand side of Eq. (12). Thus, the term*A*^{2}_{0}increases as (*r*−_{b}*r*)^{1/2}: - (ii) The wave–wave interaction between the stationary Wave 1 (
*S*^{1}_{1}) and the amplitude modulation of Wave 2 (*A*^{1}_{2}) generates the modulation of the eastward traveling Wave 1 (*A*^{2}_{1}) with the angular frequency*ω*. The amplitude |_{A}*A*^{2}_{1}| is proportional to |*S*^{1}_{1}||*A*^{1}_{2}| from the right-hand side of Eq. (13), and it increases as (*r*−_{b}*r*)^{1/2}:

*ω*are added to the periodic solution, Eqs. (28) ∼ (30), for small (

_{A}*r*−

_{b}*r*):andTo confirm this relationship, the power spectral densities for the frequency

*ω*are plotted against (

_{A}*r*−

_{b}*r*) for

*r*<

*r*on a double logarithmic chart in Fig. 9. All of the power spectral densities of (a) the zonal-mean component, (b) the amplitude of Wave 2, and (c) that of Wave 1 for the angular frequency

_{b}*ω*increase in proportion to (

_{A}*r*−

_{b}*r*) along dashed lines from the bifurcation point

*r*. These power laws support that the transition from periodic solution (

_{b}*P*) to vacillation (

*V*) is due to the Hopf bifurcation at

*r*for decreasing

_{b}*r*.

## 5. Discussion

In the previous section, our attention was focused on the parameter range of small *r* to diagnose the regime transition, in which the forced stationary Wave 1 is much smaller than that of traveling Wave 2. In the real SH stratosphere, however, the amplitude of the stationary Wave 1 is comparable to or larger than that of the traveling Wave 2. In HY04, the periodic solution for *B* = 6° and *r* = 0.08 (denoted by *P** in Table 1) was taken as an example similar to the observation. For the real atmosphere and the cases for large *r* in the present study, the periodic fluctuation of the zonal-mean zonal flow is basically explained by the energy and enstrophy exchanges with Wave 1, while it is mainly due to the exchanges with Wave 2 for small *r*. The enstrophy exchange between the mean zonal flow and Wave 1 becomes dominant for about *r* > 0.05 at *B* = 4°. As shown in Fig. 3, there are some systematic changes in the zonal-mean zonal flow and Wave 2 around *r* ∼ 0.05. The maximum value in the mean zonal flow fluctuation increases with *r* for *r* ≲ 0.05, while it is roughly constant for *r* ≳ 0.05. The variation range of Wave 2 and the amplitude of stationary Wave 2 increase for *r* ≲ 0.05, while they decrease for *r* ≳ 0.05. On the other hand, the amplitude of stationary Wave 1 and the variation range of Wave 1 increase for all ranges shown in Fig. 3.

As analyzed for the real SH stratosphere in HY04, rather periodic variations of the polar vortex due to the wave–wave interactions are often observed in late winter, and the existence of the eastward propagating Wave 2 as well as the stationary Wave 1 is the key factor that determines whether or not such variations are observed. The large activity of the eastward traveling Wave 2 is related to a barotropically unstable profile or weakly stable one of the mean polar night jet, as shown in Fig. 15b in HY04. As the season evolves from early winter to late winter in the SH, the polar night jet in the middle stratosphere becomes strong due to its shift poleward and downward (Shiotani et al. 1993). In the parameter setting for the prescribed polar night jet in the present study, *ϕ*_{0} = 55° and *B* = 4° with *r* = 0, it is barotropically stable for small *U*, while it becomes unstable to have steady-wave solutions for *U* = 180 and 210 m s^{−1} and vacillation solutions for 240 and 270 m s^{−1}, as shown in IY95. In the presence of surface topography with a finite value of *r*, stationary-wave solutions are obtained for a barotropically stable polar night jet with small *U*. The seasonal evolution of the SH middle stratosphere can be mimicked by increasing *U* with transitions from a stationary-wave solution to an irregular solution via a periodic solution. In early winter eastward traveling Wave 2 is not active because of the weak polar night jet, while it becomes active in late winter and then periodic variations of the polar vortex due to the wave–wave interaction between the traveling Wave 2 and stationary Wave 1 are observed.

As shown in Figs. 1 and 2, most of the wave disturbances are confined in high latitudes for the parameter ranges of this study, even though the fluid is assumed to fill the spherical domain. Such a situation of the confinement within the polar cap is rather similar to the rotating annulus experiment with radial differential heating (e.g., Hide and Mason 1975). Some of the laboratory experiments have been performed in order to study the wave–wave interactions between azimuthally propagating waves due to baroclinic instability and stationary waves forced by bottom topography (e.g., Li et al. 1986; Bernardet et al. 1990; Risch and Read 2005a, b, manuscripts submitted to *Geophys. Astrophys. Fluid Dyn.*). Bernardet et al. (1990) performed experiments with time-independent topography of azimuthal wavenumber 2 and obtained some examples of spatial modulation of the traveling baroclinic waves and topographically induced vacillations. Although the observations of variations of flow field in space and time were limited, the dynamical situations of these cases in the laboratory experiment may be similar to those analyzed in the present study with a low-order empirical mode expansion. The expansion into normal modes is more rigorous for the annulus experiments, as in the Lorenz (1963) work, compared to that in this paper because of the rigid side boundaries. Further observation of the flow field in space and time in this kind of laboratory experiment with bottom topography and analysis based on the mode expansion will bring deeper understanding on the wave–wave interactions that were investigated in this study.

## 6. Concluding remarks

Weakly nonlinear aspects of a barotropically unstable polar vortex in a forced–dissipative system with a zonally asymmetric surface topography were investigated with a spherical barotropic model in terms of wave–wave interactions. Two experimental parameters adopted in this study are the normalized amplitude *r* of the sinusoidal surface topography of zonal wavenumber 1, which forces a stationary planetary wave of zonal wavenumber 1 (Wave 1), and the width *B* of the prescribed zonally symmetric jet, which controls the eastward propagating Wave 2 generated by the barotropic instability. A parameter sweep experiment (summarized in Table 1) showed the predominance of periodic solution in a wide parameter space. In the periodic solutions, variations of the zonal-mean zonal flow and topographically forced Wave 1 synchronize with the periodic eastward propagation of Wave 2 through nonlinear triad interactions among these waves and the mean zonal flow. The predominance of the periodic solution is consistent with frequent observations of such rather periodic variations of the Southern Hemisphere stratospheric polar vortex due to the triad interactions (Hio and Yoden 2004).

Vacillation in the presence of surface topography is complicated because of the modulation of the zonal-mean zonal flow and wave amplitudes with the frequency of the Wave-2 phase propagation as described in section 3. Detailed investigation of the transition from the vacillation to periodic solution as *r* increases gives a deeper understanding of the periodic solutions in the presence of surface topography. Wave synchronization is explicitly shown in the polar diagrams of the complex amplitude of Waves 1 and 2 in Figs. 4 and 5. It was also diagnosed by the low-order empirical mode expansion in section 4. The dependence of the spectral powers on *r*, shown in Fig. 9, is well explained by a limited number of nonlinear triad interactions among the zonal-mean, Wave-1, and Wave-2 components, as summarized by Eqs. (28)–(30) and (34)–(36). These results are indicative of the usefulness of the low-order empirical mode expansion as a diagnosis tool. Refinement and application of such a method to study variations of the polar vortex are interesting subjects for our future work. For example, if we can construct a low-order dynamical system based on such mode expansion, the diagnosis done in section 4 will have a sound theoretical basis.

We appreciate the three anonymous reviewers for their helpful comments. The GFD-DENNOU library (SGKS Group 2001) was used for graphical output. This work was supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan, and by the Kyoto University Active Geosphere Investigations for the 21st Century COE (KAGI 21), which was approved by the MEXT of Japan.

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Regime diagram as a function of *B* and *r*. List of the symbols is given below. See text for details. The periodic solution for the parameters *B* = 6.0° and *r* = 0.08 (denoted by *P**) was investigated in details in HY04. Regime transitions depending on *r* at *B* = 4.0° (in bold) is examined in section 3b.