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4. Millennium Integrations

In the millennium integrations, statistically reliable frequency distributions of the monthly-mean polar temperature are obtained (Fig. 2). It is confirmed that interannual variability is very large during winter in the run of h0 = 1000 m (left), while it is large in spring in the run of h0 = 500 m (right). Also, the distributions are so smooth that they give information on their higher moments. The distributions in the run of h0 = 1000 m are positively skewed in autumn and bimodal in winter. On the other hand, those in the run of h0 = 500 have positive skewnesses for a longer period from autumn to spring; an extremely large skewness appears in March.

Intervals of warm winters, directly related to occurrence of SSWs, are examined in the run of h0 = 1000 m by introducing a threshold value which determines the top 200 highest winter-mean polar temperatures (indicated by the arrow in Fig. 2). Figure 3 displays frequency of intervals of the warm winters, which persist for more than t years. The cumulated frequency is surprisingly on a straight slope in the log-scale plot, as well fitted by an exponential function A exp(-t/?Ó) (denoted by the broken line). The broken line is determined by the least square method, in which ?Óis estimated at 4.5 years. The good fitting of the intervals by the exponential function means that they are described by the exponential distribution. It, in turn, indicates that the occurrence of warm winters itself is described by the Poisson distribution. Both statistical distributions mean that such warm winters occur at random from year to year. In these distributions, ?Ó represents a mean interval, and hence, on average, such warm winters take place once in 4.5 years. This nature of random occurrence holds also for the spring-mean temperature (h0 = 500 m), independently of the choice of the threshold temperature.

Dominant modes of sequence of variability through one year are extracted with the empirical orthogonal function (EOF) analysis, which is applied to the polar temperature for the 12 months from June to May. In the run of h0 = 1000 m, the most dominant mode represents a variability of the minimum temperature in winter; the minimum temperature is higher than climatology (that is, occurrence of SSWs in winter) or lower (no SSWs). Intraseasonal variability in winter (second and third modes) and final warmings in March (fourth mode) are also important to the total variance. In the run of h0 = 500 m, on the other hand, most of interannual variability is related to the timing of the seasonal march from winter to spring; the seasonal march is earlier (warmings in March or April) or later (no warmings).

A lag-correlation (regression) analysis of SSWs shows a sequence of variability in the troposphere and the stratosphere, including preconditioning and aftereffect. One month before SSWs, the zonal mean zonal wind shifts poleward and planetary waves amplify in the troposphere and the stratosphere. The aftereffect is characterized by poleward and downward propagations of anomalies of the zonal mean zonal wind and planetary-wave amplitude, which continue for several months. This feature is basically the same as the slowly-propagating anomaly of the zonal mean zonal wind (e.g., Kodera 1995).