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- Author or Editor: Shigeo Yoden x

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## Abstract

Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter.

Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solutions with different vertical structures.

Periodic solutions a series of stratospheric vacillations originally found by Holton and Mass, are obtained by time-integrations. It is found that the periodic solutions branch off from a steady solution by a Hopf bifurcation. For a finite increment of the parameter from the bifurcation point, the time average of the periodic solution is significantly different from the unstable steady solution. The nonlinear transience causes the difference.

The multiplicity of stable solutions (steady and periodic) is a possible explanation for the interannual variability of the stratosphere circulation in the middle and high latitudes during winter.

## Abstract

Nonlinear properties of a stratospheric vacillation model are investigated numerically in the light of bifurcation theory. The model is exactly the same as that used by Holton and Mass, which describes the wave-zonal flow interaction in a β-channel under a nonconservative constraint with zonal-flow forcing and wave dissipation. A set of 81 nonlinear ordinary differential equations with variables depending on time is obtained by a severe truncation and vertical differencing. All of the external parameters are fixed in time. The amplitude of the wave forcing or the intensity of zonal wind forcing at the bottom boundary is changed as a bifurcation parameter.

Three branches of the steady solutions are obtained by use of Powell's hybrid method and the pseudo-arclength continuation method. Linear stability of these solution branches is investigated by solving an eigenvalue problem in the linearized system. In some range of the bifurcation parameter, there exists a multiplicity of stable steady solutions with different vertical structures.

Periodic solutions a series of stratospheric vacillations originally found by Holton and Mass, are obtained by time-integrations. It is found that the periodic solutions branch off from a steady solution by a Hopf bifurcation. For a finite increment of the parameter from the bifurcation point, the time average of the periodic solution is significantly different from the unstable steady solution. The nonlinear transience causes the difference.

The multiplicity of stable solutions (steady and periodic) is a possible explanation for the interannual variability of the stratosphere circulation in the middle and high latitudes during winter.

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## Abstract

Dynamics of stratospheric vacillations, which were first obtained by Holton and Mass in a highly truncated spectral model, are investigated with an EP flux diagnosis based on the transformed Eulerian-mean equations. The vacillations, which are purely periodic solutions under constant external conditions, can be divided into a dynamically active period and an inactive period. Dynamics controlling the variations are different between the two periods.

During the active period the mean zonal winds vary vigorously like a stratospheric sudden warming. The variation is mainly due to the wave driving, although a large part of it is canceled by the Coriolis torque of the "residual" circulation. The transience term is important in all of the layers to determine the wave driving, while the dissipation term has a comparable magnitude in the upper layers. The wave transience is directly associated with vertical propagations of the wave-activity pulse, which is generated not at the bottom boundary, but inthe interior.

On the other hand, during the inactive period, easterly winds and negative values of the meridional gradient of the zonal mean potential vorticity appear in the lower layers within 10-30 km. These layers prevent the wave from propagating upward. The mean zonal winds above the layers are accelerated gradually by the Coriolis torque of the "residual" circulation due to the diabatic heating.

## Abstract

Dynamics of stratospheric vacillations, which were first obtained by Holton and Mass in a highly truncated spectral model, are investigated with an EP flux diagnosis based on the transformed Eulerian-mean equations. The vacillations, which are purely periodic solutions under constant external conditions, can be divided into a dynamically active period and an inactive period. Dynamics controlling the variations are different between the two periods.

During the active period the mean zonal winds vary vigorously like a stratospheric sudden warming. The variation is mainly due to the wave driving, although a large part of it is canceled by the Coriolis torque of the "residual" circulation. The transience term is important in all of the layers to determine the wave driving, while the dissipation term has a comparable magnitude in the upper layers. The wave transience is directly associated with vertical propagations of the wave-activity pulse, which is generated not at the bottom boundary, but inthe interior.

On the other hand, during the inactive period, easterly winds and negative values of the meridional gradient of the zonal mean potential vorticity appear in the lower layers within 10-30 km. These layers prevent the wave from propagating upward. The mean zonal winds above the layers are accelerated gradually by the Coriolis torque of the "residual" circulation due to the diabatic heating.

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## Abstract

A new class of vacillations is obtained in the Holton and Mass model with a different bottom boundary condition. The model is a highly truncated spectral model describing wave-zonal flow interactions in a forced-dissipative system. The mean zonal wind and the wave change their vertical structures periodically with a period of the wave progrezsion (5-10 days for the parameters used in this study). The vacillations are interpreted as an interference between a stationary wave and a topographically modified Rossby wave. The modified Rossby wave is an eigenmode of baroclinic flow in the presence of bottom topography within the framework of the highly truncated system. Time variations of the mean zonal wind are essential for the modification of the Rossby wave.

## Abstract

A new class of vacillations is obtained in the Holton and Mass model with a different bottom boundary condition. The model is a highly truncated spectral model describing wave-zonal flow interactions in a forced-dissipative system. The mean zonal wind and the wave change their vertical structures periodically with a period of the wave progrezsion (5-10 days for the parameters used in this study). The vacillations are interpreted as an interference between a stationary wave and a topographically modified Rossby wave. The modified Rossby wave is an eigenmode of baroclinic flow in the presence of bottom topography within the framework of the highly truncated system. Time variations of the mean zonal wind are essential for the modification of the Rossby wave.

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## Abstract

A simple wave-zonal flow interaction model, originally developed by Holton and Mass, is used to illustrate a rudimentary conception of seasonal and interannual variations of the stratospheric circulation.

The radiative heating is varied periodically with an annual component to investigate the response of the circulation (i.e., the seasonal variation) to the periodic forcing. The response is qualitatively different depending on the wave forcing from the troposphere. The difference resembles that of the climatological seasonal march between the Northern and the Southern hemispheres.

No example of interannual variations (namely, nonperiodic responses) was obtained for the periodic annual forcing. Interannual variation of external conditions is necessary in the present model to obtain interannual variations. A finite range of year-to-year variations of the wave forcing can produce large interannual variability as in the Northern Hemisphere.

## Abstract

A simple wave-zonal flow interaction model, originally developed by Holton and Mass, is used to illustrate a rudimentary conception of seasonal and interannual variations of the stratospheric circulation.

The radiative heating is varied periodically with an annual component to investigate the response of the circulation (i.e., the seasonal variation) to the periodic forcing. The response is qualitatively different depending on the wave forcing from the troposphere. The difference resembles that of the climatological seasonal march between the Northern and the Southern hemispheres.

No example of interannual variations (namely, nonperiodic responses) was obtained for the periodic annual forcing. Interannual variation of external conditions is necessary in the present model to obtain interannual variations. A finite range of year-to-year variations of the wave forcing can produce large interannual variability as in the Northern Hemisphere.

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## Abstract

The interaction between convectively excited waves and the mean zonal wind in the equatorial lower stratosphere is investigated with a simplified general circulation model (GCM). The model has T42 truncation, and the vertical resolution is about 700 m in the stratosphere. Although it is an “aquaplanet” model with uniform sea surface temperature, cumulus convection in low latitudes has realistic hierarchical structures with reasonable space–time spectral distributions. The model produced an oscillation having quite similar features to the equatorial quasi-biennial oscillation (QBO), although the period is 400 days.

Waves in the equatorial lower stratosphere of the model are excited mainly by the cumulus convection in low latitudes. The energy of these waves is a little larger than that observed in the real atmosphere. The dominant waves are gravity waves having an equivalent depth of about 200 m and those of 40–100 m. About half of the transport and deposition of zonal momentum contributing to the oscillation is accounted for by the gravest symmetric gravity modes: eastward momentum by Kelvin waves and westward momentum by *n* = 1 gravity waves. The momentum deposition is done over a wide range of zonal wavenumber (2–30), while about half of it is done over a period of 1–3 days. The deposition has rather continuous phase speed distributions and a considerable portion of it is provided by waves having critical levels. Since gravity waves with small intrinsic phase speeds have small vertical wavelengths, vertical grid spacings of 700 m or less appear to be required in the lower stratosphere for GCMs in order to simulate the QBO.

## Abstract

The interaction between convectively excited waves and the mean zonal wind in the equatorial lower stratosphere is investigated with a simplified general circulation model (GCM). The model has T42 truncation, and the vertical resolution is about 700 m in the stratosphere. Although it is an “aquaplanet” model with uniform sea surface temperature, cumulus convection in low latitudes has realistic hierarchical structures with reasonable space–time spectral distributions. The model produced an oscillation having quite similar features to the equatorial quasi-biennial oscillation (QBO), although the period is 400 days.

Waves in the equatorial lower stratosphere of the model are excited mainly by the cumulus convection in low latitudes. The energy of these waves is a little larger than that observed in the real atmosphere. The dominant waves are gravity waves having an equivalent depth of about 200 m and those of 40–100 m. About half of the transport and deposition of zonal momentum contributing to the oscillation is accounted for by the gravest symmetric gravity modes: eastward momentum by Kelvin waves and westward momentum by *n* = 1 gravity waves. The momentum deposition is done over a wide range of zonal wavenumber (2–30), while about half of it is done over a period of 1–3 days. The deposition has rather continuous phase speed distributions and a considerable portion of it is provided by waves having critical levels. Since gravity waves with small intrinsic phase speeds have small vertical wavelengths, vertical grid spacings of 700 m or less appear to be required in the lower stratosphere for GCMs in order to simulate the QBO.

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## Abstract

Internal variations of the troposphere–stratosphere coupled system with intraseasonal and interannual timescales are investigated in a parameter sweep experiment with a simple global circulation model under a periodic annual forcing. In order to examine the role of forced planetary waves in the variations, the amplitude of a sinusoidal surface topography is chosen as an experimental parameter; 100-yr integrations are performed for each of 10 topographic amplitudes from 0 to 3000 m.

The extratropical stratospheric circulation depends on the topographic amplitude in its mean seasonal march and intraseasonal and interannual variations. In the run without the topography, the stratospheric circulation is basically driven thermally and hardly shows interannual variation in any seasons. In the runs in which the topography is included, on the other hand, the stratospheric circulation is dynamically active and shows large interannual variation in different seasons, that is, spring for small topographic amplitudes (around 500 m) and winter for large amplitudes (around 1000 m). The mean seasonal march and interannual variation in these runs of small and large amplitudes resemble those in the Southern and Northern Hemispheres, respectively.

In this study, the annual forcing is introduced only in the stratosphere while the tropospheric condition is kept constant in time, in order to investigate downward influence from the stratosphere to the troposphere in the seasonal march. The annual response of the atmosphere can significantly penetrate into the troposphere, depending on the topographic amplitude. The downward penetration is significant for the amplitudes of planetary and synoptic-scale waves, while it is negligible for zonal mean quantities.

The empirical orthogonal function and lag correlation analyses show that a sequence of variability associated with stratospheric sudden warmings (SSWs) in the run of the topographic amplitude of 1000 m is characterized by poleward and downward propagation of anomalies of the zonal mean zonal wind and the planetary wave amplitude. Preconditioning for and the aftereffect of SSWs extend through both the stratosphere and troposphere. One month before SSWs, the polar night jet and the tropospheric jet shift poleward while planetary waves amplify in the troposphere and stratosphere. The anomalies of the zonal wind and wave amplitude further propagate poleward and downward for several months after SSWs.

## Abstract

Internal variations of the troposphere–stratosphere coupled system with intraseasonal and interannual timescales are investigated in a parameter sweep experiment with a simple global circulation model under a periodic annual forcing. In order to examine the role of forced planetary waves in the variations, the amplitude of a sinusoidal surface topography is chosen as an experimental parameter; 100-yr integrations are performed for each of 10 topographic amplitudes from 0 to 3000 m.

The extratropical stratospheric circulation depends on the topographic amplitude in its mean seasonal march and intraseasonal and interannual variations. In the run without the topography, the stratospheric circulation is basically driven thermally and hardly shows interannual variation in any seasons. In the runs in which the topography is included, on the other hand, the stratospheric circulation is dynamically active and shows large interannual variation in different seasons, that is, spring for small topographic amplitudes (around 500 m) and winter for large amplitudes (around 1000 m). The mean seasonal march and interannual variation in these runs of small and large amplitudes resemble those in the Southern and Northern Hemispheres, respectively.

In this study, the annual forcing is introduced only in the stratosphere while the tropospheric condition is kept constant in time, in order to investigate downward influence from the stratosphere to the troposphere in the seasonal march. The annual response of the atmosphere can significantly penetrate into the troposphere, depending on the topographic amplitude. The downward penetration is significant for the amplitudes of planetary and synoptic-scale waves, while it is negligible for zonal mean quantities.

The empirical orthogonal function and lag correlation analyses show that a sequence of variability associated with stratospheric sudden warmings (SSWs) in the run of the topographic amplitude of 1000 m is characterized by poleward and downward propagation of anomalies of the zonal mean zonal wind and the planetary wave amplitude. Preconditioning for and the aftereffect of SSWs extend through both the stratosphere and troposphere. One month before SSWs, the polar night jet and the tropospheric jet shift poleward while planetary waves amplify in the troposphere and stratosphere. The anomalies of the zonal wind and wave amplitude further propagate poleward and downward for several months after SSWs.

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## Abstract

A pair of millennium (1000 yr) integrations with a simple global circulation model under the same conditions as in Part I of this paper are performed for further statistical analyses on internal intraseasonal and interannual variations of the troposphere–stratosphere coupled system. According to the previous parameter sweep experiment of Part I, in which topographic amplitude *h*
_{0} is changed, the two runs of *h*
_{0} = 500 and 1000 m correspond, respectively, to the real Southern and Northern Hemispheres in stratospheric interannual variation. The 1000-yr datasets reveal detailed features of frequency distributions of stratospheric interannual variation in the two runs. The frequency distributions of the monthly mean temperature in the polar upper stratosphere are positively skewed from autumn to spring for *h*
_{0} = 500 m. On the other hand, the distributions are positively skewed in autumn and bimodal in winter for *h*
_{0} = 1000 m.

It is shown statistically that warm winters (springs) for *h*
_{0} = 1000 m (500 m) in the stratosphere related to stratospheric sudden warming (SSW) events occur at random from year to year, as described by Poisson processes. The power spectrum of the stratospheric interannual variation is close to a red noise spectrum. The multiple empirical orthogonal function (EOF) analysis applied to the polar temperature in the upper stratosphere shows different sequences of low-frequency variability through a year between the two runs. The primary EOFs for *h*
_{0} = 500 m represent variations of timing of the seasonal march from winter to spring. For *h*
_{0} = 1000 m, on the other hand, variation of the minimum temperature in winter is extracted as the leading EOF, and variations appearing in winter or early spring at shorter timescales are also significant.

A sequence of low-frequency variability associated with SSW events in January, which are defined as when the January mean temperature at the polar upper stratosphere is among the 200 highest in the 1000 yr, is examined in the framework of frequency distributions for *h*
_{0} = 1000 m. The association of variability with SSW events emerges as a bias of frequency distributions; the subset of distributions of the 200 SSW events is occasionally biased in the whole 1000-yr distributions. The bias of the preconditioning or the aftereffect in the troposphere related to SSW events is emphasized. For example, there are 130 yr in the whole 1000 yr for which planetary wave amplitude in the high-latitude troposphere is smaller than its climatology minus 1 standard deviation in February; 62 of the 130 appear one month after the 200 SSW events in January. On the basis of the present results, discussion will include (i) the difference of the randomness obtained in this study from the biennial oscillation of Scott and Haynes for stratospheric interannual variation, and (ii) implications for numerical experiments and statistical analyses on atmospheric variability and so forth.

## Abstract

A pair of millennium (1000 yr) integrations with a simple global circulation model under the same conditions as in Part I of this paper are performed for further statistical analyses on internal intraseasonal and interannual variations of the troposphere–stratosphere coupled system. According to the previous parameter sweep experiment of Part I, in which topographic amplitude *h*
_{0} is changed, the two runs of *h*
_{0} = 500 and 1000 m correspond, respectively, to the real Southern and Northern Hemispheres in stratospheric interannual variation. The 1000-yr datasets reveal detailed features of frequency distributions of stratospheric interannual variation in the two runs. The frequency distributions of the monthly mean temperature in the polar upper stratosphere are positively skewed from autumn to spring for *h*
_{0} = 500 m. On the other hand, the distributions are positively skewed in autumn and bimodal in winter for *h*
_{0} = 1000 m.

It is shown statistically that warm winters (springs) for *h*
_{0} = 1000 m (500 m) in the stratosphere related to stratospheric sudden warming (SSW) events occur at random from year to year, as described by Poisson processes. The power spectrum of the stratospheric interannual variation is close to a red noise spectrum. The multiple empirical orthogonal function (EOF) analysis applied to the polar temperature in the upper stratosphere shows different sequences of low-frequency variability through a year between the two runs. The primary EOFs for *h*
_{0} = 500 m represent variations of timing of the seasonal march from winter to spring. For *h*
_{0} = 1000 m, on the other hand, variation of the minimum temperature in winter is extracted as the leading EOF, and variations appearing in winter or early spring at shorter timescales are also significant.

A sequence of low-frequency variability associated with SSW events in January, which are defined as when the January mean temperature at the polar upper stratosphere is among the 200 highest in the 1000 yr, is examined in the framework of frequency distributions for *h*
_{0} = 1000 m. The association of variability with SSW events emerges as a bias of frequency distributions; the subset of distributions of the 200 SSW events is occasionally biased in the whole 1000-yr distributions. The bias of the preconditioning or the aftereffect in the troposphere related to SSW events is emphasized. For example, there are 130 yr in the whole 1000 yr for which planetary wave amplitude in the high-latitude troposphere is smaller than its climatology minus 1 standard deviation in February; 62 of the 130 appear one month after the 200 SSW events in January. On the basis of the present results, discussion will include (i) the difference of the randomness obtained in this study from the biennial oscillation of Scott and Haynes for stratospheric interannual variation, and (ii) implications for numerical experiments and statistical analyses on atmospheric variability and so forth.

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## Abstract

Weakly nonlinear aspects of a barotropically unstable polar vortex in a forced–dissipative system with a zonally asymmetric surface topography are investigated in order to obtain a deeper understanding of rather periodic variations of the winter circumpolar vortex in the Southern Hemisphere stratosphere that are characterized by the wave–wave interaction between the stationary planetary wave of zonal wavenumber 1 (denoted as Wave 1) and the eastward traveling Wave 2 as studied by Hio and Yoden in 2004. The authors use a spherical barotropic model with a forcing of zonally symmetric jet, dissipation, and sinusoidal surface topography. A parameter sweep experiment is performed by changing the amplitude of the surface topography, which forces the stationary Wave 1, and the width of the prescribed zonally symmetric jet, which controls the barotropic instability, to generate the traveling Wave 2. Several types of solutions from a time-independent solution to a nonperiodic irregular solution are obtained for the combination of these external parameters, but the predominant solution obtained in a wide parameter space is periodic.

Details of the wave–wave interactions are described for the transition from a quasiperiodic vacillation to a periodic solution as the increase of the amplitude of topography. Phase relationships are locked at the transition, and variations of zonal-mean zonal flow and topographically forced Wave 1 synchronize with periodic progression of Wave 2 in the periodic solution. A diagnosis with a low-order “empirical mode expansion” of the vorticity equation gives a limited number of dominant nonlinear triad interactions among the zonal-mean, Wave-1, and Wave-2 components around the transition point.

## Abstract

Weakly nonlinear aspects of a barotropically unstable polar vortex in a forced–dissipative system with a zonally asymmetric surface topography are investigated in order to obtain a deeper understanding of rather periodic variations of the winter circumpolar vortex in the Southern Hemisphere stratosphere that are characterized by the wave–wave interaction between the stationary planetary wave of zonal wavenumber 1 (denoted as Wave 1) and the eastward traveling Wave 2 as studied by Hio and Yoden in 2004. The authors use a spherical barotropic model with a forcing of zonally symmetric jet, dissipation, and sinusoidal surface topography. A parameter sweep experiment is performed by changing the amplitude of the surface topography, which forces the stationary Wave 1, and the width of the prescribed zonally symmetric jet, which controls the barotropic instability, to generate the traveling Wave 2. Several types of solutions from a time-independent solution to a nonperiodic irregular solution are obtained for the combination of these external parameters, but the predominant solution obtained in a wide parameter space is periodic.

Details of the wave–wave interactions are described for the transition from a quasiperiodic vacillation to a periodic solution as the increase of the amplitude of topography. Phase relationships are locked at the transition, and variations of zonal-mean zonal flow and topographically forced Wave 1 synchronize with periodic progression of Wave 2 in the periodic solution. A diagnosis with a low-order “empirical mode expansion” of the vorticity equation gives a limited number of dominant nonlinear triad interactions among the zonal-mean, Wave-1, and Wave-2 components around the transition point.

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## Abstract

The winter polar vortex in the Southern Hemisphere stratosphere is characterized by prominent quasi-stationary planetary waves: zonal wavenumber 1 (wave 1) and the eastward-traveling wave (wave 2). Quasi-periodic variations of the polar vortex are investigated in terms of the wave–wave interaction between wave 1 and wave 2 with both the NCEP–NCAR reanalysis dataset from 1979 to 2002 and a spherical barotropic model.

A typical case 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 amplitude. The transient wave 1 travels eastward with the same angular frequency as that of the traveling wave 2. The polar night jet also vacillates 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. The vacillation is basically due to quasi-periodic variations of the wave driven by the interference between the stationary and traveling wave 1s.

Similar periodic variations of the polar vortex are obtained in the model experiment here, in the circumstance that stationary wave 1 generated by surface topography has comparable amplitude to the eastward-traveling wave 2 that is generated by the barotropic instability of a forced mean zonal wind.

The winter polar vortex shows large interannual variability. Similar quasi-periodic variations due to wave– wave interaction often occurred for the 24 yr in late winter when the transient wave 2 was vigorous.

## Abstract

The winter polar vortex in the Southern Hemisphere stratosphere is characterized by prominent quasi-stationary planetary waves: zonal wavenumber 1 (wave 1) and the eastward-traveling wave (wave 2). Quasi-periodic variations of the polar vortex are investigated in terms of the wave–wave interaction between wave 1 and wave 2 with both the NCEP–NCAR reanalysis dataset from 1979 to 2002 and a spherical barotropic model.

A typical case 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 amplitude. The transient wave 1 travels eastward with the same angular frequency as that of the traveling wave 2. The polar night jet also vacillates 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. The vacillation is basically due to quasi-periodic variations of the wave driven by the interference between the stationary and traveling wave 1s.

Similar periodic variations of the polar vortex are obtained in the model experiment here, in the circumstance that stationary wave 1 generated by surface topography has comparable amplitude to the eastward-traveling wave 2 that is generated by the barotropic instability of a forced mean zonal wind.

The winter polar vortex shows large interannual variability. Similar quasi-periodic variations due to wave– wave interaction often occurred for the 24 yr in late winter when the transient wave 2 was vigorous.

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## Abstract

Chaotic mixing processes and transport barriers around the wintertime stratospheric polar vortex are investigated with an idealized barotropic model, previously used by Ishioka and Yoden. A barotropically unstable jet is forced in order to obtain a fluctuating polar vortex. A flow with quasiperiodic time dependence and an aperiodic flow with similar behavior are investigated using several Lagrangian methods.

A typical chaotic mixing process is observed in the quasiperiodic flow, resulting in effective mixing inside and outside of the polar vortex. The mixing regions are on the critical latitudes of several planetary waves that grow through barotropic instability. Poincaré sections give accurate locations of chaotic mixing regions, and transport barriers are identified as the edges of invariant torus regimes. In addition to the transport barriers associated with strong potential vorticity gradients, another type of transport barrier exists, which is not related to the steep potential vorticity gradient.

Chaotic mixing is dominant also in the aperiodic flow. Comparing with the quasiperiodic flow, an aperiodic flow with the same wave energy has a higher average Lyapunov exponent. This arises because the area involved in chaotic zones increases. The evolution of the correlation function is also more typical of a chaotic zone. Isolated regions are found near the center of the polar vortex, which can be explained by the invariant tori in the Poincaré sections of the quasiperiodic flow. Implications of the results for the observed “4-day wave” in the upper stratosphere are discussed.

## Abstract

Chaotic mixing processes and transport barriers around the wintertime stratospheric polar vortex are investigated with an idealized barotropic model, previously used by Ishioka and Yoden. A barotropically unstable jet is forced in order to obtain a fluctuating polar vortex. A flow with quasiperiodic time dependence and an aperiodic flow with similar behavior are investigated using several Lagrangian methods.

A typical chaotic mixing process is observed in the quasiperiodic flow, resulting in effective mixing inside and outside of the polar vortex. The mixing regions are on the critical latitudes of several planetary waves that grow through barotropic instability. Poincaré sections give accurate locations of chaotic mixing regions, and transport barriers are identified as the edges of invariant torus regimes. In addition to the transport barriers associated with strong potential vorticity gradients, another type of transport barrier exists, which is not related to the steep potential vorticity gradient.

Chaotic mixing is dominant also in the aperiodic flow. Comparing with the quasiperiodic flow, an aperiodic flow with the same wave energy has a higher average Lyapunov exponent. This arises because the area involved in chaotic zones increases. The evolution of the correlation function is also more typical of a chaotic zone. Isolated regions are found near the center of the polar vortex, which can be explained by the invariant tori in the Poincaré sections of the quasiperiodic flow. Implications of the results for the observed “4-day wave” in the upper stratosphere are discussed.