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# Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

Fund Project:

This work was jointly supported by the National Natural Science Foundation of China (Grant No. 41790474) and the National Program on Global Change and Air--Sea Interaction (GASI-IPOVAI-03 GASI-IPOVAI-06)

• The backward nonlinear local Lyapunov exponent method (BNLLE) is applied to quantify the predictability of warm and cold events in the Lorenz model. Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space. The maximum prediction lead times of each warm (cold) event on individual circles concentric with the distribution of warm (cold) regime events are roughly the same, whereas the maximum prediction lead time of events on other circles are different. Statistical results show that warm events are more predictable than cold events.
摘要: 本文基于向后非线性局部Lyapunov指数（backward nonlinear local Lyapunov exponent，BNLLE）方法，定量研究了Lorenz模型中冷暖事件的可预报性。研究结果显示，冷暖事件的最长提前预报时间在相空间中呈现明显的层状结构。在暖（冷）流型的每一圈层上，所有暖（冷）事件的最长提前预报时间基本一致，而在不同的圈层上，暖（冷）事件的最长提前预报时间则不同。基于统计结果表明，暖事件比冷事件更加容易预报。
• Figure 1.  Example of 20 error vectors superimposed on the initial state ${ x}\left({t}_{0}\right)$. The red dot represents the initial state ${ x}\left({t}_{0}\right)$, and the lines connecting the red and black dots represent the error size. The magnitude of the error vectors is 10−5.

Figure 2.  Schematic of the maximum prediction lead time (dashed line) and maximum prediction time (solid line) for state ${{x}}\left({t}_{0}\right)$ where ${t}_{0}$ is the time associated with given state ${{x}}\left({t}_{0}\right)$, t1 is the time required for the initial errors superimposed on ${{x}}\left({t}_{0}\right)$ to reach saturation, and ${t}_{-1}$ is the time associated with the corresponding initial state ${{x}}\left({t}_{-1}\right)$.

Figure 3.  Warm (red) and cold (blue) regimes projected on the xy plane of a Lorenz attractor.

Figure 4.  Spatial distributions of maximum prediction lead times for warm and cold regimes with initial error magnitudes of 10−5.

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.  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.  PDF curves of maximum prediction lead times for warm and cold events with initial error magnitudes of (a) 10−2 and (b) 10−5. Blue and red lines represent cold and warm states, respectively.

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## Manuscript History

Manuscript received: 07 April 2020
Manuscript revised: 01 June 2020
Manuscript accepted: 12 June 2020
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Quantitative Comparison of Predictabilities of Warm and Cold Events Using the Backward Nonlinear Local Lyapunov Exponent Method

###### Corresponding author: Ruiqiang DING, drq@mail.iap.ac.cn;
• 1. Department of Atmospheric and Oceanic Sciences and Institute of Atmospheric Sciences, Fudan University, Shanghai 200438, China
• 2. Key Laboratory of Physical Oceanography, Institute for Advanced Ocean Studies, Ocean University of China and Qingdao National Laboratory for Marine Science and Technology, Qingdao 266100, China
• 3. State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China
• 4. Key Laboratory of Physical Oceanography/Institute for Advanced Ocean Studies, Ocean University of China and Qingdao National Laboratory for Marine Science and Technology, Qingdao 266100, China

Abstract: The backward nonlinear local Lyapunov exponent method (BNLLE) is applied to quantify the predictability of warm and cold events in the Lorenz model. Results show that the maximum prediction lead times of warm and cold events present obvious layered structures in phase space. The maximum prediction lead times of each warm (cold) event on individual circles concentric with the distribution of warm (cold) regime events are roughly the same, whereas the maximum prediction lead time of events on other circles are different. Statistical results show that warm events are more predictable than cold events.

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