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Evans et al. (2004) classified the Lorenz attractor into warm and cold regimes, and studied when regime change occurs and the time spent in one regime. Warm and cold regimes represent warm and cold weather, respectively, in the real atmosphere. Figure 3 shows the warm and cold regimes of the Lorenz attractor projected on the x−y plane. For the warm regime, x and y are both greater than zero, whereas for the cold regime x and y are both less than zero. States corresponds to weather events. Of the 40 000 states, there are 17 292 states in the warm regime (i.e., warm events) and 18 534 states in the cold regime (i.e., cold events). The other 4174 states are in the regime transition region. In this work, we generate 10 000 initial error vectors randomly superimposed on each state x(
$ {t}_{-m} $ ), and the magnitudes of these initial error vectors are the same but in different directions. -
The initial error magnitude of the Lorenz63 model is set to 10−5. We then calculate the maximum prediction lead times for all 40 000 events. Figure 4 shows the spatial distributions of maximum prediction lead times on the Lorenz attractor. Warm events are distributed over the right wing of the Lorenz attractor and cold events are distributed over the left wing. The maximum prediction lead times of warm and cold events present obvious layered structures. The maximum prediction lead times of warm (cold) events on individual circles concentric with the distribution of warm (cold) events are roughly the same. On different circles, the maximum prediction lead times are different. In addition, we find that the maximum prediction lead times of warm and cold events are similar overall, with small differences. In this work, the parameter r is 28 which is larger than 1. So the Lorenz attractor has three unstable stationary points (Mukougawa et al., 1991, Mu et al., 2002). One unstable stationary point is the origin (0, 0, 0). The other two unstable stationary points are located on (
$ \sqrt{\beta (r-1)},\sqrt{\beta (r-1)},r-1 $ ) and ($ -\sqrt{\beta (r-1)},-\sqrt{\beta (r-1)},r-1 $ ), which are the centers of warm and cold regimes, respectively. The warm (cold) events on an individual orbit are circled around the unstable stationary point on the warm (cold) regime. In our opinion, the properties of all the events on an individual circle may be the same, indicating similar predictabilities of these events. So, the maximum prediction lead times of events on an individual circle are similar. Nese (1989) pointed out that the predictabilities of states vary with the phase space of the Lorenz attractor. Therefore, the predictabilities of events vary with different circles. Taking account of the two factors—the same properties of events on an individual circle and predictabilities varying with circles—the maximum prediction lead times of warm and cold events present obvious layered structures. -
To further investigate which type of event is more predictable, we apply statistical information for the maximum prediction lead times of warm and cold events (Table 1). Figure 5 is a boxplot of maximum prediction lead times for warm and cold events. The largest of the maximum prediction lead times of the 17 292 warm events is slightly lower than that of the 18 534 cold events. The other four statistical variables [the first quartile (Q1), median value (Q2), the third quartile (Q3), and minimum value] of warm events are all higher than those of the cold events, indicating that warm events are more predictable than cold events.
Minimum Q1 Median Q3 Maximum Warm 10.99 12.94 13.61 14.24 16.19 Cold 10.02 12.42 13.29 14.02 16.42 Table 1. Statistical information for the maximum prediction lead times of warm and cold events.
Figure 5. Boxplot of the maximum prediction lead time of warm and cold events with initial error magnitudes of 10−5. Red solid lines indicate the median value (Q2). The bottoms and tops of the boxes denote the first quartile (Q1) and third quartile (Q3), respectively. The lower and upper solid horizontal lines represent the minimum value (Q1 − 1.5IQR) and maximum value (Q3 + 1.5IQR) of the maximum prediction lead times of warm (cold) events, respectively, and IQR = Q3 − Q1.
Figure 6 shows probability histograms of maximum prediction lead times of the two types of event under two scenarios with different initial error magnitudes. The maximum prediction lead times of warm and cold events both form Gaussian distributions. Extreme warm and cold events occur with low frequency, and thus extreme maximum prediction lead times are of low probability. For non-extreme events, the probabilities of maximum prediction lead times for warm events are generally higher than those of cold events.
Figure 6. Probability histograms for maximum prediction lead times of (a, c) cold and (b, d) warm events with initial error magnitudes of (a, b) 10−2 and (c, d) 10−5.
Figure 7 shows probability distribution function (PDF) curves of maximum prediction lead times for warm and cold events. For both magnitudes of initial error, the probability distributions of maximum prediction lead times for warm events are shifted to longer times compared with those for cold events. The maximum prediction lead times of warm events are thus greater than those of cold events with the same probability. This demonstrates that warm events are more predictable than cold events.
Minimum | Q1 | Median | Q3 | Maximum | |
Warm | 10.99 | 12.94 | 13.61 | 14.24 | 16.19 |
Cold | 10.02 | 12.42 | 13.29 | 14.02 | 16.42 |