## Abstract

Internal variability of a troposphere–stratosphere coupled system is investigated in a series of numerical experiments with a simple global circulation model under a perpetual winter condition. In order to examine the relative importance of forced planetary waves in the interaction with the zonal mean zonal flow and baroclinic disturbances, amplitude of a sinusoidal surface topography of zonal wavenumber 1 or 2 is swept as an experimental parameter; 1000-day integrations are performed for 110 combinations of the external parameters.

Intraseasonal variability of the extratropical winter stratosphere in the parameter sweep experiment of the zonal wavenumber-1 topography is classified into four regimes, dependent on the topographic amplitude: (I) a nearly radiative equilibrium state; (II) small undulations of the polar vortex; (III) intermittent breakdowns of the polar vortex, or occurrence of stratospheric sudden warming events; and (IV) a usually weak and warm state of the polar vortex. The behavior of planetary waves in the stratosphere is characterized by linear propagation in regime I, and by quasi-linear, or weakly nonlinear, wave–mean flow interaction in regime II. In regimes III and IV, on the other hand, it is in a highly nonlinear state, reflecting the occurrence of sudden warming events and the usually distorted polar vortex, respectively. The Northern Hemisphere winter stratosphere corresponds to regime III in the intraseasonal variations, while the southern counterpart is close to regime II.

A lag-correlation analysis shows that the dynamical linkage between the stratosphere and the troposphere is also dependent on the regimes. The vertical linkage is primarily the upward control in regime I; planetary waves are generated by nonlinear interaction of baroclinic disturbances in the troposphere and propagate into the stratosphere. In regime II, stratospheric variations are largely confined in the stratosphere, although planetary waves are generated in the troposphere. The linkage in regime III, on the other hand, is inevitably two-way. Planetary wave variability has upward influence from the troposphere to the stratosphere as well as the zonal mean zonal wind shows preconditioning in the troposphere before stratospheric sudden warming events. Downward propagation of signals to the upper troposphere are also seen in the zonal mean temperature in high latitudes; it is higher than normal at the final stage of the sequence of sudden warming events. The linkage is also two-way in regime IV.

## 1. Introduction

The extratropical stratosphere exhibits considerable variations with year-to-year timescales as well as with month-to-month scales (e.g., Randel 1992). The interannual and intraseasonal variations are especially large in the Northern Hemisphere (NH) winter (e.g., Labitzke 1982). Such variations are largely due to the occurrence of stratospheric sudden warming (SSW), or breakdown of the polar vortex; a major SSW occurs only once in 2 or 3 yr on average and its timing is at random within the winter season. An averaged temperature in the polar stratosphere over a winter or a month is much higher than normal, if SSW events take place during that period.

It has been widely accepted that such stratospheric variations are primarily caused by variations of tropospheric circulation. For example, Matsuno (1971) assumed impulsive intensification of tropospheric planetary waves in his pioneering work on the theory of SSWs, in order to initiate wave-induced deceleration of the zonal mean westerlies in the stratosphere. In his numerical result, transient planetary waves, which propagated from the troposphere, caused an SSW subsequently, because they have a negative (westward) force on the zonal mean zonal flow at their leading edge due to the “wave transience effect.” Such upward influence from the troposphere prior to SSW events has been confirmed by case studies of some SSW events in the real atmosphere and by numerical experiments with general circulation models (GCMs) (see Andrews et al. 1987, section 6.2 for the observational analyses and section 11.3 for the GCM experiments).

Another possible cause of the stratospheric variations is an internal process of the extratropical stratosphere due to nonlinear interactions. A simple wave–mean flow interaction model shows substantial internal variability known as stratospheric vacillation under a time-constant lower boundary condition placed at the tropopause (Holton and Mass 1976). The vacillation is characterized by the periodic occurrence of SSWs with an interval of several months (Yoden 1987a,b). Such intraseasonal variability under a time-constant lower boundary condition has been obtained in some hierarchy of stratosphere only models; a wave–mean flow interaction model in spherical geometry (Scott and Haynes 1998) and a modified GCM with fixed troposphere (Christiansen 1999). The Holton–Mass model does not bear interannual variability but has a purely periodic response when the zonal mean zonal flow is forced periodically with an annual cycle (Yoden 1990), because the extratropical channel model completely loses the memory of the previous winter in summer. In a model with more degrees of freedom in spherical geometry, year-to-year variations can be obtained for a periodic annual forcing of the zonal mean zonal flow (Scott and Haynes 1998). They pointed out that the timescale for relaxation of zonal mean zonal angular momentum in low latitudes, contrary to that for thermal relaxation, is long enough to retain such memory of one winter till the next and to cause such an interannual variation. They called such a character “the low-latitude flywheel.”

The above two groups of studies made very opposite assumptions on the dynamical linkage between the troposphere and the stratosphere. The former relationship assumes “slaving stratospheric variations”; the prescribed tropospheric variations induce stratospheric variations but the stratospheric variations have no effects on the troposphere (or, the lower boundary condition). The latter assumes “independent stratospheric variations”; only the internal variations of the stratosphere are permitted in the models on the assumption of time-constant lower boundary condition. These two extreme assumptions are useful in theoretical studies to idealize the problem of the linkage from different viewpoints. In the real atmosphere, however, the interaction between the troposphere and the stratosphere may exist and play a nonnegligible role in the variations with intraseasonal and interannual timescales; if it were negligible, the stratosphere only models stated above would have more sound foundations.

Some observational studies have been done to investigate the dynamical linkage between the troposphere and the stratosphere on the intraseasonal and interannual timescales for NH winter (e.g., Baldwin et al. 1994; Kodera et al. 1996; Thompson and Wallace 1998). By analyzing monthly or seasonal mean data, these studies showed that zonally symmetric seesaw patterns of geopotential height in the stratosphere, alternating between the polar region and midlatitudes, are strongly coupled to similar patterns of geopotential height in the troposphere and sea level pressure. The patterns in the troposphere and at the surface resemble the North Atlantic oscillation, but with more zonal symmetry. The deep signature of the polar-vortex oscillation was named the Arctic oscillation (AO) by Thompson and Wallace (1998), which has been recognized as a dominant and robust mode of the stratosphere–troposphere coupled variability.

There are some recent studies that claim downward propagation of the AO signature from the stratosphere to the troposphere. Kuroda and Kodera (1999) showed that anomalies of the zonal mean zonal wind, which shift poleward and downward from the stratosphere to the troposphere, together with anomalies of the convergence of Eliassen–Palm flux of planetary waves, correspond to the appearance of the AO pattern in the 500-hPa geopotential height field. By using a GCM, Christiansen (2000) also showed the connection of zonal mean zonal wind anomalies in the stratosphere to the AO in the troposphere. Baldwin and Dunkerton (1999) examined time evolutions as well as vertical structures of the AO to find that the AO anomalies typically appear in the stratosphere at first and propagate downward to the surface. As for the downward influence with shorter timescales within 1 month, Kodera and Chiba (1995) made a case study of the 1984–85 SSW event, and notified changes in the tropospheric circulation associated with the stratospheric event.

Downward influence of SSWs to the troposphere can be seen in GCM experiments too. Yoden et al. (1999) analyzed a 7200-day dataset of the Berlin Troposphere–Stratosphere–Mesosphere GCM under a perpetual January integration, which is more suitable to study the intraseasonal variability. Their composite analysis of 64 SSW events shows some downward influences of SSWs to the troposphere, which are different depending on the dominant zonal wavenumber during each event. For wavenumber-1 type of SSWs, the zonal mean temperature in the polar region of the upper troposphere and lower stratosphere remains lower than normal after the events, and the zonal mean zonal wind in mid- and high latitudes is also stronger. For wavenumber-2 type of SSWs, on the other hand, signals of the SSWs descend down to the upper troposphere after the events for about 1 month.

These observational and GCM studies, which show some evidences of downward influences to the troposphere as well as the upward control from the troposphere, limit the justification of the assumption of slaving stratospheric variations or independent stratospheric variations, or rather, these studies raise a new question how deeply the stratosphere and the troposphere are coupled with each other.

In order to understand the fundamental dynamics of the coupled system, mechanistic circulation models (MCMs) are useful in which large-scale dynamical processes are explicitly described while some physical processes are simplified, because the real atmosphere and GCMs are so complicated due to various physical processes they include. Employing such an MCM, Scinocca and Haynes (1998) investigated planetary wave variability in the stratosphere generated by nonlinear interactions of baroclinic eddies in the troposphere. In their framework, however, the downward influence is not so significant, because drastic variations in the stratosphere such as SSWs are absent by the lack of forced planetary waves, or due to zonally symmetric boundary condition at the surface. Forced planetary waves have a crucial role in the dynamical coupling between the troposphere and the stratosphere.

In this study, vertical dynamical coupling in the presence of forced planetary waves is investigated with a troposphere–stratosphere MCM; a sinusoidal surface topography is introduced, while any longitudinal thermal contrast is not incorporated into the model to avoid the contamination of two different effects. In total, 110 runs of 1000-day integrations are performed by changing the topographic amplitude and some other external parameters. The purpose of this study is to clarify dynamical nature of the troposphere–stratosphere coupled system by showing qualitative dependence of stratospheric variability and its relation to tropospheric variability on the topographic amplitude, or forced planetary waves. Some basic physical quantities such as zonal means and wave amplitudes are examined to characterize such variability in the stratosphere and the troposphere.

This kind of parameter sweep experiment is useful to study a system in which nonlinear processes play an important role. It is rather common for nonlinear systems to have a large qualitative change of time-dependent behavior for a small change of an external parameter. Yoden (1987a) did such a parameter sweep for the Holton–Mass model, the system of which has only 81 degrees of freedom, to show the nonlinear nature of the stratospheric vacillations. Recent progress of computer resources enables us to make a parameter sweep study with an MCM that has full dynamical process with *O*(10^{5}) degrees of freedom.

We explore the dynamical coupling between the troposphere and the stratosphere that is highly dependent on the nature of SSWs generated in the model for each topographic amplitude. The present MCM with no surface topography is basically the same as the situation studied by Scinocca and Haynes (1998); only the upward control is expected mainly through planetary waves that are generated by the modulation of baroclinic waves. If the surface topography is included into the model, on the other hand, stationary planetary waves are generated by the interaction between the surface topography, the zonal mean zonal flow, and baroclinic waves in the troposphere. The interaction is sensitive to the amplitude of the topography, and realistic SSWs take place in a certain range of the amplitude.

The present paper is organized as follows. The present MCM is described in section 2 and its performance is shown in section 3 for the control run, in which several SSW events take place. Results of the parameter sweep are documented in section 4 about time-mean states, time variations and correlations of some basic quantities. Discussion is in section 5 and conclusions in section 6.

## 2. Model

The model employed in this study is a *σ*-coordinate spectral primitive equation model on the globe, which is based on the AGCM5 (Swamp Project 1998). The horizontal resolution is about 5.6° × 5.6° with a triangular truncation at total wavenumber 21. The vertical resolution is about 2.7 km in the stratosphere with 42 *σ* levels from the surface up to the mesopause. Time integration is performed with Δ*t* = 20 min for 1200 days in each run from the initial state of an isothermal atmosphere (240 K) at rest with small disturbances. The first 200 days are discarded for the present analysis to avoid the initial transience.

Some physical processes in the model are simplified to investigate fundamental dynamics of the troposphere–stratosphere coupled system. Short- and longwave radiations are represented by a Newtonian heating and cooling term relaxed toward a prescribed basic temperature *T**(*ϕ*, *z*), which is displayed in Fig. 1a. Here *ϕ* is latitude and *z* is height. The profile mimics a radiative equilibrium state in NH winter and is held constant in time.

The standard basic temperature profile *T**(*ϕ*, *z*) comprises tropospheric part *T*^{*}_{t}(*ϕ*, *z*) and stratospheric one *T*^{*}_{s}(*ϕ*, *z*) with smooth connection between them. The tropospheric part *T*^{*}_{t}(*ϕ*, *z*) is identical to the basic temperature profile used by Akahori and Yoden (1997), which is symmetric with respect to the equator. The stratospheric part *T*^{*}_{s}(*ϕ*, *z*) is calculated from the basic zonally symmetric zonal wind profile *U*^{*}_{s}(*ϕ*, *z*) of Scott and Haynes (1998), by requiring gradient wind balance [Andrews et al. 1987, Eq. (3.4.1c)] and by assuming *T*^{*}_{s}(*ϕ* = 0°, *z* = 0 km) = 250 K and a stepwise vertical profile of the static stability over the equator; *N*^{2} (*z*) = 1.3 × 10^{−4} s^{−2} for *z* ≤ 12 km, 5.0 × 10^{−4} s^{−2} for 12 km < *z* < 50 km, and 2.5 × 10^{−4} s^{−2} for 50 km ≤ *z*. Since *σ* coordinates are employed in the model, *T**(*ϕ*, *z*) is converted into *T*^{*}_{σ}(*λ*, *ϕ*, *σ*) in the computation, where *λ* is longitude. While *T**(*ϕ*, *z*) is independent of *λ*, *T*^{*}_{σ}(*λ*, *ϕ*, *σ*) depends on *λ* if a surface topography exists, because the surface pressure is dependent on the topographic height. When *z* is converted into *σ*, a time-independent standard field of surface pressure, which is obtained by assuming hydrostatic balance of an isothermal atmosphere (240 K), is used instead of a time-dependent one. The radiative relaxation rate is given as a function of height, *α*(*z*) = {1.5 + tanh[(*z* − 35 km)/7 km]} × 10^{−6} s^{−1}, as in Holton and Mass (1976), which is also converted into *α*_{σ}(*λ*, *ϕ*, *σ*) in the computation.

The model has dry atmosphere with no moist processes. All the processes related to water vapor such as moist convection are excluded. Radiative effects by clouds can not be incorporated in the Newtonian heating and cooling scheme.

Dry convective adjustment is retained in the model. Internal horizontal dissipation in the form of ∇^{4} is applied to the temperature, vorticity and divergence fields with a damping time of 2 days for the maximum wavenumber 21. The boundary layer is simply parameterized by the Rayleigh friction at the bottom level with a relaxation time of 0.5 days. The Rayleigh friction is also used in sponge layers above 50 km with a coefficient of *β*(*z*) = {1.02 − exp[(50 km − *z*)/40 km]} × 5 × 10^{−6} s^{−1} as in Scott and Haynes (1998).

A hypothetical surface topography as in Yamane and Yoden (1998) is used in the form of

where *μ* = sin*ϕ* and *m* is zonal wavenumber. The topography consists of the single zonal wavenumber, to avoid contamination by forced waves of different zonal wavenumbers. The topographic amplitude *h*_{0} is chosen as an experimental parameter, and is swept from 0 m to 3000 m with *m* fixed at 1 or 2; 50 runs are performed for *m* = 1 (noted as W1) changing *h*_{0}, and 16 runs are done for *m* = 2 (W2). In order to examine robustness of obtained results, 28 runs for *m* = 1 and 16 runs for *m* = 2 are further carried out with another basic temperature profile, which is constructed by modifying the constants over the equator. The whole stratosphere is warmer by about 20 −30 K in this profile than in the standard one, while the troposphere is identical. In total, 110 runs of 1000-day integrations are done in the present study. The parameter sweep with successive small increment of *h*_{0} is very necessary, because the interactions between the zonal mean zonal flow, forced planetary waves, and baroclinic waves in the troposphere are highly nonlinear so that drastic change in the time-dependent behaviors does exist for a small change of *h*_{0}.

## 3. Control run

In order to examine the model performance, climatological time mean and variance fields of zonal mean quantities and planetary waves are described for the run of *h*_{0} = 1000 m and *m* = 1 with the standard basic temperature. We regard this run as the control experiment, because some realistic SSWs are simulated in the run. Case studies of several SSW events in the control run are also done to show the model performance of time variations.

### a. Climatology

Figure 1 displays the 1000-day mean and variance fields of the zonal mean temperature and the zonal mean zonal wind in the control run as well as the basic temperature profile *T** used in Newtonian heating and cooling. The 1000-day mean of the zonal mean temperature field [*T*] (Fig. 1b) exhibits basic aspects in the real atmosphere (e.g., Holton 1992, Fig. 12.2) such as strong negative gradient from the equator to the Poles in the troposphere and monotonical decrease from the summer South Pole to the winter North Pole (NP) in the stratosphere. Here square brackets denote the zonal mean and an overbar does the time mean. The departure of [*T*] from *T** (Fig. 1d) shows that [*T*] is much higher than *T** in the mid- and high latitudes of the winter stratosphere and lower in the tropical stratosphere; such Newtonian cooling in the winter stratosphere is balanced by adiabatic heating due to wave-induced downward motions. In the troposphere, [*T*] is higher than *T** in high latitudes while lower in low latitudes, showing that baroclinic waves reduce the equator–Pole temperature gradient of the basic state. The 1000-day mean of the zonal mean zonal wind [*U*], displayed in Fig. 1e, basically satisfies a thermal wind balance with [*T*]. The westerly subtropical jet in the both hemispheres, the westerly polar night jet in the NH extratropical stratosphere, and the easterly jet in the Southern Hemisphere (SH) stratosphere agree qualitatively with those observed in the atmosphere (e.g., Holton 1992, Figs. 12.2 and 12.3). The standard deviation of the daily fields of the zonal mean temperature from the 1000-day mean, *σ*_{[T]}, is large in NH high latitudes and has its maxima over the NP in the middle and upper stratosphere and in the mesosphere (Fig. 1c). The standard deviation of [*U*], *σ*_{[U]}, peaks in the high-latitude (*ϕ* ∼ 60°N) stratopause (Fig. 1f).

Several properties of planetary waves of geopotential height are examined in Fig. 2 for zonal wavenumber-1 and -2 components (hereafter referred to as wave 1 and wave 2, respectively). The amplitude |(*Z*)_{1}| and phase *θ*_{(Z)1} of the stationary component of wave 1 are displayed in Figs. 2a,b, respectively. Here, stationary wave is defined as wave component in the 1000-day mean geopotential height field. They are similar to those in NH winter of the real atmosphere (e.g., Andrews et al. 1987, Fig. 5.4); they show a general feature of wave propagation, up into the high-latitude stratosphere and toward the equator. The 1000-day mean of the daily amplitude of wave 1 |*Z*_{1}| (Fig. 2c), which is contributed by both of the stationary and traveling components, shows a similar pattern as |(*Z*)_{1}| but the value is larger by about 10%–40% in the stratosphere, indicating that the stationary component is dominant in wave 1. The standard deviation of the daily wave-1 amplitude *σ*_{|Z1|} has the maximum in the extratropical upper stratosphere (Fig. 2d), which is similar to *σ*_{[U]}.

The four quantities for wave 2 are also displayed in Figs. 2e–h. The stationary wave-2 amplitude (Fig. 2e) is much less, by a factor of about 1/3, than the mean daily amplitude of wave 2 (Fig. 2g), which indicates that the traveling component is dominant in wave 2. The standard deviation of wave-2 *σ*_{|Z2|} peaks in the extratropical upper stratosphere (Fig. 2h), as *σ*_{|Z1|}. In each quantity (except wave phase), wave 1 is largest in the stratosphere among all the wave components, indicating the dominant role of wave 1 in stratospheric variability in this system.

### b. Time variations

Figure 3 displays time–height sections of some dynamical quantities for a period of 150 days in the control run, during which typical SSW episodes take place; (Fig. 3a) [*T*] − [*T*] in the polar region of *ϕ* = 86°N, (Fig. 3b) [*U*], and (Fig. 3c) |*Z*_{1}|, both of which are averaged over latitudes 40°N ≤ *ϕ* ≤ 80°N. The latitudinal average is applied to [*U*] and |*Z*_{1}| because their large values fluctuate with time in the latitudes. The extratropical stratosphere remains in a relatively undisturbed period until about day 340; the zonal mean temperature in the polar region is below the time mean (Fig. 3a) and the zonal mean westerly wind is strong in mid- and high latitudes (Fig. 3b). Thereafter the extratropical stratosphere enters into a relatively disturbed period, in which three SSW episodes take place in succession around days 355, 380, and 400. The polar temperature shows rapid warmings over 60 K within several days, accompanied by deceleration of the zonal mean westerly wind in the stratosphere. In the last event, the mean zonal wind is replaced by the easterlies even in the middle stratosphere (*p* ∼ 10 hPa). After the disturbed period, the stratosphere radiatively returns to another undisturbed period. Even as long as 1 month after the last SSW event, the extratropical stratosphere is warmer and the westerlies are weaker than in the former undisturbed period. This indicates that longer time is necessary for the reestablishment of the polar vortex.

Each of the SSWs is preceded by planetary wave amplification in the upper stratosphere around days 345, 375, and 395 (Fig. 3c). The amplifications in the stratosphere exhibit different ways of vertical coupling, although |*Z*_{1}| is not the best measure of wave activity. The first amplification can be traced back down to the surface, which seems to be an example of the upward influence; wave 1 amplifies in the troposphere and propagates to the stratosphere. Such upward coupling, however, is not clearly seen in the other two amplification episodes, suggesting importance of internal variability in the stratosphere. The three amplifications of |*Z*_{1}|, as well as many others, exhibit downward propagation of local maxima (denoted by dots in the figure) in the stratosphere and some of the propagation penetrate down to the troposphere. It looks anomalous that |*Z*_{1}| shows downward propagation from the stratosphere around day 350 and remains large in the troposphere over the following 2 weeks or so. This suggests a possibility that the tropospheric planetary waves are influenced by the first SSW episode.

Polar stereographic charts of geopotential height on the particular four days 340, 370, 400, and 430 (denoted by vertical lines in Fig. 3) are displayed in Fig. 4 for three pressure levels of 1.2, 57, and 549 hPa. On day 340 in the former undisturbed period, the polar vortex in the stratosphere is strong and rather zonally symmetric around the NP without large amplitude planetary waves. The tropospheric circulation on that day is characterized by the shift of the polar vortex off from the NP and by a wave breaking pattern near *λ* = 0°; the corresponding amplification of |*Z*_{1}| persists for a week or so and propagates up to the stratosphere to cause the first SSW event (Fig. 3). On day 370, which is about the beginning of the second SSW episode, the polar vortex is weak and off from the NP at *p* = 1.2 hPa. The polar vortex in the troposphere is slightly off from the NP, indicating that |*Z*_{1}| becomes smaller than on day 340. By day 400, which is at the peak of the third SSW event, a dramatic change has occurred in the upper stratosphere; at *p* = 1.2 hPa, the polar vortex is most weakened and shifted off from the NP, while an anticyclone dominates the polar region. The polar vortex at *p* = 57 hPa is also highly elongated. On day 430 after the SSW event, the polar vortex in the stratosphere is in the process of radiative recovery.

## 4. Parameter sweep

Results of the parameter sweep experiments, mainly for the series of W1 with the standard basic temperature, are described from viewpoints of 1000-day mean states, time variations in the stratosphere, and correlations between the stratosphere and troposphere.

### a. Time mean states

Figure 5 displays latitude–height sections of the 1000-day mean of the zonal mean zonal wind, [*U*], in 26 runs of the standard W1 series. The mean state of the polar night jet depends on *h*_{0} in its magnitude and location. For the runs of small *h*_{0} (0 m ≤ *h*_{0} ≤ 400 m), the polar night jet is very strong, over 110 m s^{−1}, and its location gradually shifts poleward with *h*_{0} from *ϕ* ∼ 50°N to *ϕ* ∼ 60°N. For the runs of larger *h*_{0}, the polar night jet is more strongly dependent on *h*_{0}; the maximum speed decreases from 110 m s^{−1} in the run of *h*_{0} = 400 m to 60 m s^{−1} in the run of *h*_{0} = 800 m, while the location shifts poleward and downward from *ϕ* = 60°N and *p* = 1 hPa to *ϕ* = 65°N and *p* = 2 hPa. For the runs of 900 m ≤ *h*_{0} ≤ 2200 m, the maximum speed and location are not very dependent on *h*_{0}, about 40 ∼ 60 m s^{−1} near *ϕ* = 65°N and *p* = 2 hPa. As *h*_{0} is further increased up to *h*_{0} = 3000 m, the polar night jet weakens to 20 m s^{−1}. Also note that [*U*] near *ϕ* = 30°N and *p* = 1 hPa increases with *h*_{0} for 2200 m ≤ *h*_{0}. The subtropical jet in the troposphere is not so dependent on *h*_{0}.

Such dependence of the 1000-day mean states on *h*_{0} is summarized in Fig. 6, in which several quantities at various levels are plotted as functions of *h*_{0} for all the 50 runs of the standard W1 series. The *h*_{0} dependence of the magnitude of the polar night jet, mentioned in Fig. 5, is confirmed in Fig. 6a (upper panel), which plots the maximum of the relative zonal mean zonal angular momentum Max([*U*] cos*ϕ*) in 0°N ≤ *ϕ* ≤ 90°N; the maximum zonal angular momentum is large in 0 m ≤ *h*_{0} ≲ 200 m, decreases with *h*_{0} in 200 m ≲ *h*_{0} ≲ 800 m, and is only weakly dependent on *h*_{0} for larger *h*_{0}. It is noticeable that the angular momentum above the middle stratosphere (*p* ≤ 12 hPa) decreases in 2000 m ≲ *h*_{0}.

The mean temperature [*T*] in the polar stratosphere (Fig. 6b, upper panel) and the maximum of the stationary wave-1 amplitude Max(|(*Z*)_{1}|) in the stratosphere (Fig. 6c, upper panel) have *h*_{0} dependences corresponding to those of the maximum zonal angular momentum. The polar temperature is low in 0 m ≤ *h*_{0} ≲ 400 m, increases with *h*_{0} in 400 m ≲ *h*_{0} ≲ 800 m, and is almost independent of *h*_{0} for larger *h*_{0}. In 2000 m ≲ *h*_{0}, the polar temperature slightly increases except at *p* = 1.2 hPa. The maximum wave-1 amplitude increases with *h*_{0} in 0 m ≤ *h*_{0} ≲ 600 m and is roughly constant for larger *h*_{0}. The increase of wave-1 amplitude above the middle stratosphere (*p* ≤ 12 hPa) is sharpest around *h*_{0} = 400 m. The wave-1 amplitude in the upper stratosphere (*p* = 1.2 and 2.6 hPa) decreases in 2000 m ≲ *h*_{0}.

In the troposphere, the zonal mean quantities are much less dependent on *h*_{0} (Figs. 6a,b, lower panels), and their dependences are rather different from those in the stratosphere. The maximum angular momentum decreases with *h*_{0} in 0 m ≤ *h*_{0} ≲ 500 m, but increases for larger *h*_{0} except at *p* = 875 hPa. The polar temperature becomes slightly higher in 0 m ≤ *h*_{0} ≲ 1500 m except at *p* = 875 hPa while nearly independent for larger *h*_{0}. The maximum wave-1 amplitude in the troposphere (Fig. 6c, lower panel) becomes larger with *h*_{0}, sharply in 0 m ≤ *h*_{0} ≲ 500 m and gradually in 500 m ≲ *h*_{0}. It therefore follows that the stationary wave-1 amplitude linearly increases with *h*_{0} only for 0 m ≤ *h*_{0} ≲ 500 m both in the troposphere and stratosphere.

Because synoptic-scale waves are important especially for the tropospheric circulation, wave 5 is examined as a measure of the baroclinic disturbances. Daily amplitude of wave 5 |*Z*_{5}| (Fig. 6d) below the lower stratosphere (*p* ≤ 57 hPa) decreases with *h*_{0} in 0 m ≤ *h*_{0} ≲ 500 m, increases in 500 m ≲ *h*_{0} ≲ 1600 m, and decreases again for larger *h*_{0} up to *h*_{0} = 3000 m. Synoptic-scale waves of other zonal wavenumbers, such as wave 4 and wave 6, also have similar *h*_{0} dependence (not displayed).

### b. Time variations in the stratosphere

Figure 7 displays 1000-day time series of the zonal mean temperature [*T*] near the polar stratopause for the 26 runs of the standard W1 series, which is a measure of the dynamical condition of the polar vortex. For the runs of small *h*_{0} (0 m ≤ *h*_{0} ≤ 300 m), the polar temperature is always low, indicating a steady and strong polar vortex. As *h*_{0} is increased to *h*_{0} = 400 m, there appear small fluctuations characterized by highly intermittent, sporadic warming of several tens kelvins. The following range of 500 m ≲ *h*_{0} ≲ 2000 m has a distinguishing feature of intermittent occurrence of SSWs; after a preceding cold state, the polar temperature shows rapid increase and large fluctuations, and returns to the next cold state. The frequency of such SSW events is about 5–10 times for the 1000 days, and therefore the timescale of one SSW event is about 100–200 days. For larger *h*_{0} up to *h*_{0} = 3000 m, the temperature exhibits frequent warmings and seldom comes back to the cold state.

In order to examine the *h*_{0} dependence of the time series of [*T*] in Fig. 7 from a statistical viewpoint, their histograms are displayed in Fig. 8a with grayscale plots. Similar histograms for [*U*] and |*Z*_{1}| in the extratropical middle stratosphere and for |*Z*_{5}| in the extratropical upper troposphere are also displayed in Figs. 8b–d. Since [*U*], |*Z*_{1}|, and |*Z*_{5}| have large variations in the extratropical latitudes, the same latitudinal average in 40°N ≤ *ϕ*_{0} ≤ 80°N as in Figs. 3b,c is applied to obtain the histograms. Based on these histograms, all the runs of the W1 series are subjectively classified into four regimes that are qualitatively different from each other in stratospheric time variations. One possible classification is (I) 0 m ≤ *h*_{0} ≤ 350 m, (II) 375 m ≤ *h*_{0} ≤ 675 m, (III) 700 m ≤ *h*_{0} ≤ 1900 m, and (IV) 2000 m ≤ *h*_{0} ≤ 3000 m, denoted by vertical lines in the figure.

In regime I, [*T*], [*U*], and |*Z*_{1}| show little variance; [*T*] remains low, [*U*] is always strong westerly, and |*Z*_{1}| is small. As *h*_{0} is increased in the regime II, the time means of these three quantities change largely reflecting that the probability distributions of [*T*] and [*U*] become highly skewed one. For example, the time mean of [*T*] increases with *h*_{0} and its distribution is skewed with high probability toward the cold side. The distributions of these three quantities show large variances in both of regimes III and IV. The distinction between regimes III and IV is based on the larger skewnesses of [*T*] and [*U*] in regime III, which can be confirmed intuitively by looking at the time series in Fig. 7; contrast between the cold and warm periods is clearer in regime III due to the intermittent occurrence of SSWs.

The frequency distribution of |*Z*_{5}| in the upper troposphere (Fig. 8d) is not very dependent on the regimes, but has a little skewness to the small-amplitude side in the whole range of *h*_{0}.

A similar classification is also possible for another series of W1 experiment in which the warmer profile is used (not displayed). This indicates that these classifications generally hold independent of the basic temperature profile.

The classification into the four regimes dependent on *h*_{0} in the standard W1 series is also supported by the relationship between the zonal mean zonal wind and planetary wave amplitude. Figure 9 displays two-dimensional scatter diagrams with respect to the 1000-day data of [*U*] and |*Z*_{1}| in the extratropical middle stratosphere for the 26 runs of the standard W1 series. Owing to the large number of samples (*N* = 1000), the correlation coefficient *r* (written in each panel multiplied by a factor of 100) has strong statistical significance even if |*r*| is not so close to unity. In regime I, the diagrams show highly dense distributions with large [*U*] and small |*Z*_{1}|, which indicates that the polar vortex is always strong with little planetary wave activity. The correlation coefficient is close to zero. The scatter diagrams in regime II show tight distributions along straight slopes, and the distributions have strong negative correlations equal to or below −0.7; the zonal mean zonal wind is relatively weak when the wave-1 amplitude is large, and vice versa, although the variance of the mean zonal wind is not large. In regime III, the variable ranges of the mean zonal wind become wide due to intermittent, large departures from the tight distributions similar to those in regime II. Such intermittent excursions, which are directly related to SSW events shown in Fig. 7, are presented as anticlockwise orbits on the [*U*] − |*Z*_{1}| phase plane; after |*Z*_{1}| extremely amplifies, [*U*] decreases to small values, or to negative ones occasionally, and then [*U*] gradually recovers to the strong westerly state with decaying |*Z*_{1}|. Negative correlations in regime III are not so strong as those in regime II, due to the large departures. In regime IV, such excursions become more frequent, and the tight distribution on the plane that has strong negative correlation disappears.

### c. Lag correlations between the stratosphere and the troposphere

Vertical linkage between the stratosphere and the troposphere in time variations is investigated with a lag-correlation analysis. Figure 10a displays a time lag–height section of the correlation coefficient of the zonal mean temperature at a given latitude *ϕ*_{1} = 86°N in the run of *h*_{0} = 0 m of the standard W1 series; the lag correlation coefficient is calculated between the time series of [*T*] at a reference point of *ϕ*_{0} = 86°N and *p*_{0} = 2.6 hPa (denoted by × in the figure) and those with a time lag *lag* at all other pressure levels *p* at the same latitude *ϕ*_{0}, denoted by *r*_{[T]} (*lag*, *ϕ*_{1} = *ϕ*_{0}, *p*; *ϕ*_{0} = 86°N, *p*_{0} = 2.6 hPa). It is noted again that the abundance of the data (*N* = 900) contributes to the strong statistical significance of the correlation coefficients. The zonal mean temperature in the polar region shows positive correlation in the stratosphere, while negative one in the troposphere and the mesosphere. The largest correlation coefficient is found near *lag* = 0 day at each pressure level; the large coefficient such as *r* > 0.6 is confined within |*lag*| ≲ 5 days. The coefficient becomes smaller with either increasing or decreasing *lag* at each level.

The lag correlation of the zonal mean zonal wind in the extratropics (*ϕ*_{1} = 58°N) with the zonal mean zonal wind averaged over latitudes 40°N ≤ *ϕ*_{0} ≤ 80°N at *p*_{0} = 12 hPa, *r*_{[U]} (*lag*, *ϕ*_{1} = 58°N, *p*; 40°N ≤ *ϕ*_{0} ≤ 80°N, *p*_{0} = 12 hPa), is displayed in Fig. 10b. Note again that the key time series are averaged latitudinally to capture large variations of [*U*] in the extratropics. As the time series at *ϕ*_{1} = 58°N significantly contribute the key time series averaged in 40°N ≤ *ϕ*_{0} ≤ 80°N, the correlation is close to unity near the reference level around *lag* = 0 day. The strong correlation spreads for the 2 months around *lag* = 0 day from the surface to the mesosphere. The correlation exceeds 0.5 even 1 month before and after *lag* = 0 day at each level. It indicates that the zonal mean zonal wind varies very gradually in the extratropics.

The lag correlation of wave-1 amplitude in the extratropics, *r*_{|Z1|} (*lag*, *ϕ*_{1} = 58°N, *p*; 40°N ≤ *ϕ*_{0} ≤ 80°N, *p*_{0} = 12 hPa), shows strong correlation between the stratosphere and the troposphere, confined around *lag* = 0 day (Fig. 10c). The time lag of the maximum correlation at each level is nearly *lag* = 0 day in the stratosphere and several days before *lag* = 0 day near the surface. The backward tilt of the maximum correlation with decreasing height is indicative of the upward propagation of wave-1 component generated in the troposphere. The lag correlation of wave-5 amplitude, *r*_{|Z5|} (*lag*, *ϕ*_{1} = 53°N, *p*; 40°N ≤ *ϕ*_{0} ≤ 80°N, *p*_{0} = 254 hPa), also shows strong correlation in the stratosphere and the troposphere, although the time lag of the maximum correlation is hardly seen above the middle troposphere (Fig. 10d). This is an indication of the evanescent nature of synoptic-scale waves in the stratosphere.

Vertical sections of the lag correlations in the run of *h*_{0} = 1000 m (Fig. 11) exhibit very different features from those in the run of *h*_{0} = 0 m, due to the influence of forced planetary waves, or the intermittent occurrence of SSWs. The lag correlations are calculated in the same way as in Fig. 10. The dominant feature of *r*_{[T]} is a downward propagation; the height of the maximum correlation at each *lag* (denoted by closed circles in the figure) decreases slowly with time through the 2 months around *lag* = 0 day from the lower mesosphere (*p* ∼ 0.7 hPa) to the lower stratosphere (*p* ∼ 70 hPa). The maximum correlation at *lag* = 10 days is seen near *p* ∼ 10 hPa with *r* ∼ 0.7, and further down at *lag* = 20 days near *p* ∼ 30 hPa with *r* ∼ 0.6. The correlation exceeds 0.5 even in the lower stratosphere at *lag* = 30 days. A further inspection reveals another fast descent of the timing of the maximum correlation at each pressure level (denoted by open circles) around *lag* = 0 day; the timing is *lag* ∼ −1 day at *p* ∼ 1 hPa while *lag* ∼ 1 day at *p* ∼ 10 hPa.

These descents of the maximum correlation of [*T*] indicate two kinds of downward propagation of the temperature variations associated with SSW events, because large fluctuations of the temperature during SSW events significantly contribute to the large correlation coefficient. Such descents can be directly identified in the time–height section of temperature deviation displayed in Fig. 3a. The fast descent reflects the positive anomalies with a timescale of several days in each of the three SSW events (around days 355, 380, and 400), while the slow one with a timescale of a couple of months is closely related to the sequence of the three SSW events. The first SSW episode takes place near the stratopause (*p* ∼ 1 hPa), and the second one does in somewhat lower altitudes near *p* ∼ 3 hPa about a month later. The third episode has large fluctuation in much lower altitudes around *p* ∼ 10 hPa. This kind of sequence of SSW events are seen several times in the time series of the polar temperature during the whole 1000-day period in Fig. 7 (around days 500 and 1100). The lag correlation captures a common sequence of such time variations.

Each of the lag correlations of the zonal mean zonal wind *r*_{[U]} and wave-1 amplitude *r*_{|Z1|} also shows the two kinds of the maximum correlation descent (Figs. 11b,c). For example, the slow descent of *r*_{[U]} extends from *p* ∼ 2 hPa at *lag* = −15 days to *p* ∼ 40 hPa at *lag* = 30 days, even when *r*_{[U]} is as large as 0.6. The fast one extends from the upper stratosphere at *lag* = −2 or −3 days to the lower stratosphere (*p* ∼ 30 hPa) at *lag* = 1 or 2 days. The two descents of the maximum correlations of *r*_{[U]} and *r*_{|Z1|} are also identified in the time–height sections of [*U*] and |*Z*_{1}| (Figs. 3b,c); the fast descent is seen in each of the westerly decelerations, and the slow one is also seen as the sequence of the fast descents.

Wave-1 amplitude has also positive correlation over 0.4 in the middle and upper troposphere, which leads by about 1 week. It indicates a correlation of wave-1 amplification between the troposphere and the stratosphere preceding SSW events, such as the case on about day 340 in Fig. 3c. The positive correlation in the troposphere, however, is isolated from that around the reference point in the middle stratosphere due to a node of the local minimum just below *p* = 100 hPa. It suggests a complex way how planetary waves are generated in the troposphere to propagate to the stratosphere in the presence of surface topography; such upward propagation of wave 1 as the case around day 340 is not so usual. Positive correlation of wave-5 amplitude *r*_{|Z5|} is more confined below the lower stratosphere in this run (Fig. 11d).

It is worthwhile to mention that no significant correlation is obtained between planetary-wave (waves 1, 2) amplitudes in the stratosphere and synoptic-wave (waves 4, 5, 6) amplitudes in the troposphere in either runs of *h*_{0} = 0 m and 1000 m (not displayed). This suggests that the interactions between the planetary waves and baroclinic disturbances are very complicated without any simple linear relationship; wave amplitude may not be a good measure to diagnose the interactions in the troposphere.

The lag–height sections of the lag correlations (Figs. 10 and 11) are useful to characterize the vertical coupling. One way to show changes of the vertical coupling in the whole range of *h*_{0} is to repeat such sections for each run, but it would be very tedious. Instead, *h*_{0} dependence of the vertical coupling is examined by extracting important information of the lag correlations by the procedure explained below. First, the lag-correlation coefficients *r*_{X} (*lag*, *ϕ*, *p*; *ϕ*_{0}, *p*_{0}; *h*_{0}) of the four quantities (denoted by *X*) are calculated for −50 days ≤ *lag* ≤ 50 days at all meridional points of 0°N ≤ *ϕ* ≤ 90°N and 10^{−1} hPa ≤ *p* ≤ 10^{3} hPa in each run of 0 m ≤ *h*_{0} ≤ 3000 m. The same key time series are used as in Figs. 10 and 11. Then, the largest absolute value of *r*_{X}, denoted by *R*_{X} with the sign, is searched at each pressure level *p* in each run of *h*_{0}. Thus, *R*_{X} is obtained as a function of *p* and *h*_{0}, together with the lag and latitude that give *R*_{X} (denoted by *Lag*_{X} and Φ_{X}, respectively). Figure 12 displays *R*_{X}, *Lag*_{X} and Φ_{X} of the four quantities as functions of *p* and *h*_{0}, in which only values of |*R*_{X}| ≥ 0.5 are shown. The figure shows that the vertical correlation is also different, dependent on *h*_{0}, or the four regimes separated by the vertical lines.

In regime I, the zonal mean temperature (Fig. 12a) shows positive *R*_{[T]}, which extends from the reference level in the upper stratosphere (denoted by horizontal line in the figure) down to the lower stratosphere (*p* ∼ 100 hPa). It is accompanied by *Lag*_{[T]} close to zero and Φ_{[T]} in high latitudes (Φ_{[T]} ≥ 70°N). Positive correlation of the zonal mean zonal wind spreads from the mesosphere down to the surface (Fig. 12b). The lag *Lag*_{[U]} is close to *Lag*_{[U]} = 0 day and the latitude Φ_{[U]} is upright in the extratropics. The closeness of *Lag*_{[T]} and *Lag*_{[U]} to zero confirms the appearance of their maximum correlations near *lag* = 0 day (Figs. 10a,b). Wave-1 amplitude has positive *R*_{|Z1|} in the stratosphere (1 hPa ≲ *p* ≲ 100 hPa) with *Lag*_{|Z1|} leading a few days with decreasing height (Fig. 12c), which indicates the upward propagation of wave-1 component. Positive correlation of wave-5 amplitude penetrates from the troposphere up to the upper stratosphere with *Lag*_{|Z5|} ∼ 0 day (Fig. 12d).

As *h*_{0} is increased to regime II, strong correlations of [*U*] and |*Z*_{1}| are confined in the stratosphere; upward propagation of wave-1 amplitude as in regime I disappears in regime II. Positive correlation of |*Z*_{5}| becomes confined in the troposphere with increasing *h*_{0}.

In regime III, the descent of *R*_{[T]} and *R*_{|Z1|} is confirmed. The slow descent of *R*_{[T]} is particularly remarkable; positive *R*_{[T]} extends down to the upper troposphere (*p* ∼ 200 hPa), and *Lag*_{[T]} is larger than 10 days while Φ[*T*] is directed toward low latitudes with decreasing height (Φ_{[T]} < 60°N in the upper troposphere). These features confirm the fact that the zonal mean temperature in the high-latitude upper troposphere is higher at the end of the sequence of SSWs than the time average (Figs. 3a and 11a). The zonal mean zonal wind has positive *R*_{[U]} from the surface to the mesosphere. It is accompanied by positive *Lag*_{[U]} of 1 or 2 days in the middle stratosphere (*p* ∼ 30 hPa), while by negative *Lag*_{[U]} below the lower stratosphere, which leads by more than 20 days in the midlatitude troposphere (40°N ≤ Φ_{[U]} ≤ 50°N). Since the mean westerly wind in the stratosphere is largely decelerated in SSW events and contribute to the strong correlation, these features of [*U*] is related to SSW events; for example, the positive *R*_{[U]} with the negative *Lag*_{[U]} in the midlatitude troposphere indicates that the zonal mean westerly wind weakens there before SSWs. The lag correlation of [*U*] has a signal of the slow descent of *r*_{[U]} in the lower stratosphere (Fig. 11b), but it is not the maximum at the levels and does not appear in Fig. 12b. Wave-1 amplitude also shows positive *R*_{|Z1|} down to the lower stratosphere (*p* ∼ 100 hPa), where *Lag*_{|Z1|} is 1 or 2 days. Positive correlation of wave-5 amplitude is confined below the lower stratosphere with little time lag.

As *h*_{0} is further increased in regime IV, the strong correlation of [*T*] becomes confined above the middle stratosphere. The zonal mean zonal wind does not have strong correlation in the troposphere as in regime III, suggesting that the vertical coupling is different in time variations of [*U*], particularly before SSWs, between regimes III and IV.

## 5. Discussion

Responses of the stratosphere to topographically forced planetary waves and its coupling to the troposphere were investigated in this study. The intraseasonal variations of the troposphere–stratosphere coupled system in winter are classified into the four regimes depending on the topographic amplitude *h*_{0}.

The extratropical stratospheric circulation is in almost linear state in regimes I and II, although the tropospheric circulation exhibits highly nonlinear evolution. In regime I, nonlinear interaction of baroclinic disturbances in the troposphere generates planetary wave of zonal wavenumber 1 (and also 2), as in the situation studied by Scinocca and Haynes (1998). The wave-1 amplitude increases in proportion to *h*_{0} both in the troposphere and the stratosphere (Figs. 6c and 8c), but is so small that the wave 1 does not interact significantly with the zonal mean zonal wind in the stratosphere (Fig. 9); the wave 1 only propagates linearly in the stratosphere (Figs. 10c and 12c). As a result, the polar vortex is close to the thermally driven state with little time variations, independent of *h*_{0}; the zonal mean temperature near the polar stratopause is always cold and close to the radiative equilibrium (Figs. 7 and 8a), and the zonal mean zonal wind in the extratropical stratosphere is strong westerly (Fig. 8b).

In regime II, wave 1 obtains large amplitude enough to interact with the zonal mean zonal wind in the stratosphere (Fig. 9). In spite of the large amplitude of wave 1, the interaction between the zonal mean zonal wind and wave 1 does not still show highly nonlinear behavior but does roughly linear relationship. The fluctuation of the polar temperature seen in Fig. 7 is a direct result of the interaction. Thus, this regime II can be regarded as a quasi-linear regime. As *h*_{0} is increased in regime II, wave 1 amplitude becomes larger and the wave–mean flow interaction is enhanced (Fig. 9); if wave 1 has larger amplitude in the stratosphere with increasing *h*_{0}, it decelerates the polar night jet more largely, and the decelerated jet, in turn, allows more upward propagation of wave 1. This positive feedback results in the sharp changes of the stratospheric time mean states with *h*_{0} in this regime (Figs. 6a–c).

It is observed only in regime I that wave 1 linearly propagates from the troposphere to the stratosphere as seen in Fig. 10c. The linear propagation of wave 1 disappears in the other regimes, which is revealed in similar sections of the lag correlation of wave 1 amplitude for each run to Figs. 10 and 11 (not displayed). This suggests that the generation process of planetary waves is complicated in the presence of topography.

On the contrary to the linear regime I and the quasi-linear regime II, the following wide ranges of regimes III and IV are highly nonlinear regimes. The zonal mean zonal wind and wave 1 amplitude in the middle stratosphere do not show any linear relationships (Fig. 9), as in regime II. In regime III, the planetary wave amplitude on occasions reaches large values enough to intermittently cause drastic reductions of the zonal mean westerly wind. That is, such breakdown of the polar vortex, or SSW event is a highly nonlinear response in the winter stratosphere. The cause of the intermittent occurrence of SSWs may be intermittent planetary wave forcing from the troposphere. While the polar vortex breaks down intermittently in regime III, it is always weak and warm in the regime IV, as seen in Fig. 7. The occurrence of SSWs results in quite different dependences on *h*_{0} from those in regimes I and II.

The vertical coupling between the stratosphere and the troposphere is two-way in regimes III and IV. Time variations of forced planetary waves in the stratosphere is positively correlated to those in the troposphere with a time delay of about 1 week (Fig. 11c). However, the strong correlation in the troposphere is not continuously connected to that in the stratosphere, but there is a node of weak correlation near the tropopause. The correlation gap poses a question about the conventional idea of the forcing mechanism of planetary waves by flow over topography in the linear theory [e.g., Holton 1992, Eq. (7.95)]. The gap rather suggests the planetary waves that propagate into the stratosphere are generated in the whole depth of the troposphere by the interaction processes among the zonal mean zonal wind, forced planetary waves and baroclinic disturbances. The generation mechanism of planetary waves in the presence of topography with some finite amplitude still remains an open question.

Downward influence is also significant in regime III in which SSWs take place. The zonal mean temperature, the zonal mean zonal wind and planetary wave amplitude exhibit the fast and slow downward propagations of their time variations (Figs. 11a–c and 12a–c), which are largely associated with SSWs. The fast one reflects their time variations in each SSW event in the upper and middle stratosphere with a timescale of several days. It is traditionally considered that the descent corresponds to that of the critical level in each SSW. On the other hand, the slow one is dominated by the sequence of SSW events; the first SSW event occurs in the upper stratosphere, with the subsequent events in lower altitudes down to the middle stratosphere. Even in the troposphere, there is a discernible signal in the extratropics following the sequence of SSWs, with the zonal mean temperature higher than normal after the sequence.

The slow propagation might be related to the slowly propagating anomaly of the zonal mean zonal wind (e.g., Kodera 1995; Kuroda and Kodera 1999); in NH winter, an anomaly of the zonal mean zonal wind slowly propagates from the subtropical upper stratosphere to the polar region of the lower stratosphere for 2 or 3 months. The present study emphasized that the slow downward propagation is a result of the sequence of SSW events, while Kodera et al. (2000) proposed a “conditioning” of the slowly varying field for a major SSW, which subsequently occurs only once.

It is worthwhile relating the four regimes found in this study with a troposphere–stratosphere coupled model to the regimes in Scott and Haynes (2000) with a stratosphere-only model. It is qualitatively common that the extratropical stratosphere is dynamically quiet for small *h*_{0} (time-constant wave forcing near the tropopause in the stratosphere-only model) and that it becomes active with increasing *h*_{0}. But there is also a significant difference in the dynamically active regimes; the regimes exhibit rather regular variability, such as stratospheric vacillations, in the stratosphere-only model, while very irregular variability (regimes II–IV) in the troposphere–stratosphere model. The irregularity of stratospheric variability in the present model is largely influenced by the time-varying tropospheric circulation. Figure 13 shows irregular variations of amplitude and also latitudinal structure of wave 1 in the lower stratosphere in the three runs of *h*_{0} = 0, 500, and 1000 m. It seems that a time-constant wave amplitude near the tropopause is not a good assumption at least for the present results.

Yoden (1987a) pointed out the possibility of multiple stable states, a cold undisturbed polar vortex, and a warm disturbed one, of the winter stratosphere for some parameter ranges, using a highly truncated simple wave–mean flow interaction model (Holton and Mass 1976). The existence of multiple stable states can lead to hysteresis in the simplified model, when the wave forcing near the tropopause is changed in time. The existence of hysteresis is suggested also in the present model from another series of experiment in which *h*_{0} is changed in time very gradually. There are some differences on transitions of dynamical conditions of the stratosphere between the two sets of runs in which *h*_{0} is increased or decreased in time (not displayed).

In this study, stratospheric variability and its coupling to tropospheric one are described based on some naive quantities such as zonal mean temperature and zonal wind. It is therefore a necessary work in future to make a dynamical analysis on the stratospheric variability and the vertical coupling. One main open question is on the nature of the tropospheric circulation, or the generation mechanism of vertically propagating planetary waves, which is related to the present classification of regimes in the stratospheric variability. It is also interesting to describe the troposphere–stratosphere coupled variability in the parameter sweep experiment from a viewpoint of the AO (Thompson and Wallace 1998); qualitative changes of the AO and its coupling to the stratospheric variability with forced planetary waves can be explored in our framework of simplified MCM.

If we remember intraseasonal variability in the real atmosphere (e.g., Labitazke 1982; Randel 1992), the SH winter stratosphere, where dramatic SSWs are hardly observed, corresponds to regime II, while the NH counterpart, where minor SSWs occur repeatedly and major SSWs do every 2 or 3 yr, corresponds to regime III. These correspondences are consistent with that the zonal asymmetry of the lower boundary conditions in planetary scales is much larger in NH than in SH; planetary waves generated by the zonal asymmetry are important to cause such dramatic variability in the NH stratosphere through the interaction with the zonal mean zonal wind. The parameter sweep experiment reveals that the correlations between the stratosphere and troposphere are dependent on the regimes (Fig. 12). This result suggests that the vertical coupling in the real atmosphere associated with intraseasonal variability in winter should be different between NH and SH. Further investigations are necessary on the vertical couplings in both hemispheres from a common viewpoint.

## 6. Conclusions

Dynamical coupling between the troposphere and the stratosphere in the intraseasonal timescales was investigated in a series of numerical experiments under a perpetual winter condition with an idealized global circulation model, which explicitly describes interaction among the zonal mean zonal flow, forced planetary waves, and baroclinic disturbances. In order to examine the relative importance of forced planetary waves in the interaction, the amplitude, *h*_{0}, of a sinusoidal surface topography of zonal wavenumber 1 or 2 (*m* = 1 or 2) was chosen as an experimental parameter and swept from 0 to 3000 m. For each combination of the parameters, 1000-day integrations were performed with either of two basic temperature profiles for Newtonian thermal forcing. In total, 110 runs were carried out in this study.

A case study of several stratospheric sudden warming (SSW) events, simulated in a run of *h*_{0} = 1000 m and *m* = 1, showed two ways how the stratospheric variability is coupled to the tropospheric one; not only was the upward propagation of enhanced planetary waves confirmed before SSW events, but also downward influence after SSWs was observed.

Intraseasonal variability of the extratropical winter stratosphere obtained in the parameter sweep experiment with *m* = 1 was classified into four regimes, depending on *h*_{0}: (I) 0 m ≤ *h*_{0} ≤ 350 m, in which the polar vortex is always strong and cold, close to the radiative equilibrium; (II) 375 m ≤ *h*_{0} ≤ 675 m, in which the polar vortex slightly undulates; (III) 700 m ≤ *h*_{0} ≤ 1900 m, in which the polar vortex intermittently breaks down, causing SSW events; and (IV) 2000 m ≤ *h*_{0} ≤ 3000 m, in which the polar vortex is weak and warm at all times. The stratospheric wave behavior is linear in regime I, while “quasi-linear” (i.e., a typical wave–mean flow interaction occurs) in regime II. The following wide range of regime III is characterized by a highly nonlinear behavior of stratospheric waves reflecting the occurrence of SSWs. The winter stratosphere in regime III alternates between dynamically active and inactive periods with a timescale of *O*(10^{2}) days. Several SSWs occur in succession during the active periods, while radiative relaxation to the radiative equilibrium state dominates in the inactive periods. The regime IV also shows highly nonlinear behavior. The time variations in regime II are like those in the SH winter stratosphere, while the variations in regime III are like those in the NH counterpart.

It was shown in a lag-correlation analysis that the dynamical coupling between the troposphere and the stratosphere is also different dependent on the regimes. In regime I where stratospheric variability is very small, the vertical coupling is symbolically expressed as the “slaving stratospheric variations” controlled by the troposphere; planetary-scale variability in the stratosphere is caused by nonlinear interaction of baroclinic disturbances in the troposphere.

In regime II, strong correlations associated with the stratospheric variations are confined in the stratosphere (i.e., “independent stratospheric variations”) suggesting internal variability of the stratosphere, although the planetary waves are generated in the troposphere.

In regimes III and IV, on the other hand, the dynamical coupling is inevitably two-way. During dramatic variability in the stratosphere such as SSW events, the troposphere influences the stratosphere and vice versa. Upward influence can be seen in positive correlation of planetary wave amplitude in the middle stratosphere to that in the troposphere with a time lag of about one week. Downward influence is also significant in the zonal mean temperature, associated with the sequence of SSWs, which appear at first in the upper stratosphere to eventually reach the middle stratosphere. The sequence of SSWs exerts an influence even on the troposphere at its final stage, when the zonal mean temperature in the high-latitude upper troposphere is higher than normal. Stratospheric variability is so closely coupled to the troposphere that the independent stratospheric variations are unlikely in these regimes.

## Acknowledgments

The present graphic tools were based on the codes in the GFD-DENNOU Library (SGKS Group 1999). Calculations were performed on VPP500 and VPP800 of the Kyoto University Data Processing Center. This work was supported in part by the Grant-in-Aid for Scientific Research of the Ministry of Education, Culture, Sports, Science, and Technology of Japan and by the Grant-in-Aid for the Research for the Future Program “Computational Science and Engineering” of the Japan Society for the Promotion of Science.

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## Footnotes

*Corresponding author address:* Dr. Shigeo Yoden, Department of Geophysics, Kyoto University, Kyoto 606-8502, Japan. Email: yoden@kugi.kyoto-u.ac.jp