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Mar.  2023

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# Impact of Perturbation Schemes on the Ensemble Prediction in a Coupled Lorenz Model

• Based on a simple coupled Lorenz model, we investigate how to assess a suitable initial perturbation scheme for ensemble forecasting in a multiscale system involving slow dynamics and fast dynamics. Four initial perturbation approaches are used in the ensemble forecasting experiments: the random perturbation (RP), the bred vector (BV), the ensemble transform Kalman filter (ETKF), and the nonlinear local Lyapunov vector (NLLV) methods. Results show that, regardless of the method used, the ensemble averages behave indistinguishably from the control forecasts during the first few time steps. Due to different error growth in different time-scale systems, the ensemble averages perform better than the control forecast after very short lead times in a fast subsystem but after a relatively long period of time in a slow subsystem. Due to the coupled dynamic processes, the addition of perturbations to fast variables or to slow variables can contribute to an improvement in the forecasting skill for fast variables and slow variables. Regarding the initial perturbation approaches, the NLLVs show higher forecasting skill than the BVs or RPs overall. The NLLVs and ETKFs had nearly equivalent prediction skill, but NLLVs performed best by a narrow margin. In particular, when adding perturbations to slow variables, the independent perturbations (NLLVs and ETKFs) perform much better in ensemble prediction. These results are simply implied in a real coupled air–sea model. For the prediction of oceanic variables, using independent perturbations (NLLVs) and adding perturbations to oceanic variables are expected to result in better performance in the ensemble prediction.
摘要: 本文基于一个简单的耦合Lorenz模型，探讨了多尺度模式的集合预报初始扰动构造相关问题。集合预报试验使用了四种初始扰动方法：随机扰动（RP）、繁殖向量（BV）、集合变换卡尔曼滤波（ETKF）和非线性局部 Lyapunov 向量（NLLV）方法。结果表明，无论使用哪种方法，预报的初始阶段，集合平均与控制预报相近。耦合Lorenz模型由慢系统和快系统耦合而成。由于误差在不同时间尺度系统呈现不同的增长模态，快系统经过较短的一段时间后，集合平均的结果开始优于控制预报，然而，慢系统经过相对较长的时间后，集合预报才开始起作用。此外，由于不同尺度之间的相互反馈过程，无论是对快变量还是对慢变量叠加扰动，都有助于提高慢系统和快系统的预报技巧。对不同初始集合扰动生成方法进行比较，发现NLLVs总体上优于BVs和 RPs，NLLVs和ETKFs的预报能力几乎相当。当向慢变量叠加扰动时，独立扰动（NLLVs和ETKFs）在集合预报中表现出更好的预报技巧。将简单模型的结果引申到真实的海气耦合模式。我们推测，对于海洋变量的预报，使用独立扰动（NLLVs）并且在海洋变量叠加扰动，会取得更好的集合预报效果。
• Figure 1.  Time evolution of variables for the coupled Lorenz model: (a) slow variables and (b) fast variables.

Figure 2.  Projections of the coupled Lorenz model on three two-dimensional planes: (a)–(c) for the slow variables and (d)–(f) for the fast variables.

Figure 3.  Schematic diagram of the generation of NLLVs [adapted from Hou et al. (2018)]. The creation of NLLV1 is similar to the creation of BV. To acquire the NLLV2, a pair of RPs is initially added to the analysis state. The evolved perturbations (grey dashed line) are orthogonalized with the NLLV1 (blue dashed line) to produce the NLLV2 (green dashed line) using a Gram–Schmidt re-orthonormalization (GSR) procedure. Similarly, the vectors NLLVn are orthogonalized with NLLV1, NLLV2, NLLV3, …, NLLVn–1.

Figure 4.  Illustration of the initialization and forecasting procedure. Numbers represent the integration steps, and 1 step = 0.005 tus.

Figure 5.  Panels (a)–(c) Evolution of control forecasts (light blue) against the true state (light red) as a function of lead time for (a) the whole system, (b) the fast subsystem, and (c) the slow subsystem (in the Euclidean norm). (d) Mean growth rate in the form of Lyapunov exponent (value × 100) of 10000 samples as a function of lead time from the coupled Lorenz model for the control run [the whole system (light purple), fast subsystem (light orange), and slow subsystem (light blue)].

Figure 6.  Mean RMSE (solid lines) and ensemble spread (dashed lines) of 10 000 samples as a function of lead time for the control run (black), RP method (red), BV method (blue), ETKF method (purple), and NLLV method (green) after adding perturbations to all variables: (a) the whole system, (b) the fast subsystem, and (c) the slow subsystem.

Figure 7.  Mean RMSE (solid lines) and ensemble spread (dashed lines) of 10 000 samples in the fast subsystem as a function of lead time for the control run (black), random perturbation method (red), BV method (blue), ETKF method (purple), and NLLV method (green) after adding perturbations to different variables: (a) adding perturbations to both fast variables and slow variables, (b) adding perturbations only to fast variables, and (c) adding perturbations only to slow variables.

Figure 8.  Mean RMSE (solid lines) and ensemble spread (dashed lines) of 10 000 samples in the slow subsystem as a function of lead time for the control run (black), random perturbation method (red), BV method (blue), ETKF method (purple), and NLLV method (green) after adding perturbations to different variables: (a) adding perturbations to both fast variables and slow variables, (b) adding perturbations only to fast variables, and (c) adding perturbations only to slow variables.

Figure 9.  Panels (a)–(c) RMSE of 10 000 samples based on NLLV and BV methods at (a) 3 tus, (b) 6 tus, and (c) 9 tus in the slow subsystem. The upper right-hand corner indicates the ratio of samples where RMSE for the NLLV method is smaller than the RMSE for the BV method in (a)–(c). Panels (d)–(f) are the same as (a)–(c), but for an ensemble spread of 10 000 samples. The upper right-hand corner indicates the ratio of samples where the ensemble spread for the NLLV method is larger than for the BV method in (d)–(f). Panels (a)–(f) are based on the experiments which add perturbations to both fast and slow variables.

Figure 10.  (a) Basic Brier score (BS) for the event ${\phi _1}$ (${\phi _1}$: where $X_3^{({\rm{f}})}$ is the climatological mean to the distance of one standard deviation) of ensemble forecasts based on NLLVs (green line), ETKFs (purple line), BVs (blue line), and RPs (red line) as a function of lead time. Panel (b) is the same as (a), but for event ${\phi _2}$ (${\phi _2}$: where $X_3^{({\rm{s}})}$ is the climatological mean to the distance of one standard deviation).

Figure 11.  The histogram of the Talagrand distribution for different member intervals. The horizontal dashed lines denote the expected probability for the ensemble forecasts based on (a) BVs, (b) NLLVs, and (c) ETKFs at 2 tus. Panels (a)–(c) are based on the experiment which adds perturbations to both fast and slow variables and predicts the variable $X_3^{({\rm{f}})}$. Panels (d)–(f) are the same as (a)–(c), but at 6 tus and predicted variable is $X_3^{({\rm{s}})}$.

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## Manuscript History

Manuscript revised: 19 May 2022
Manuscript accepted: 01 June 2022
###### 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

## Impact of Perturbation Schemes on the Ensemble Prediction in a Coupled Lorenz Model

###### Corresponding author: Ruiqiang DING, drq@bnu.edu.cn;
• 1. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics (LASG), Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029, China
• 2. College of Earth Science, University of Chinese Academy of Sciences, Beijing 100049, China
• 3. State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing 100875, China
• 4. Frontiers Science Center for Deep Ocean Multispheres and Earth System (FDOMES)/Key Laboratory of Physical Oceanography/Institute for Advanced Ocean Studies, Ocean University of China, Qingdao 266100, China
• 5. Laboratory for Ocean Dynamics and Climate, Pilot Qingdao National Laboratory for Marine Science and Technology, Qingdao 266237, China
• 6. Department of Atmospheric and Oceanic Sciences and Institute of Atmospheric Sciences, Fudan University, Shanghai 200438, China

Abstract: Based on a simple coupled Lorenz model, we investigate how to assess a suitable initial perturbation scheme for ensemble forecasting in a multiscale system involving slow dynamics and fast dynamics. Four initial perturbation approaches are used in the ensemble forecasting experiments: the random perturbation (RP), the bred vector (BV), the ensemble transform Kalman filter (ETKF), and the nonlinear local Lyapunov vector (NLLV) methods. Results show that, regardless of the method used, the ensemble averages behave indistinguishably from the control forecasts during the first few time steps. Due to different error growth in different time-scale systems, the ensemble averages perform better than the control forecast after very short lead times in a fast subsystem but after a relatively long period of time in a slow subsystem. Due to the coupled dynamic processes, the addition of perturbations to fast variables or to slow variables can contribute to an improvement in the forecasting skill for fast variables and slow variables. Regarding the initial perturbation approaches, the NLLVs show higher forecasting skill than the BVs or RPs overall. The NLLVs and ETKFs had nearly equivalent prediction skill, but NLLVs performed best by a narrow margin. In particular, when adding perturbations to slow variables, the independent perturbations (NLLVs and ETKFs) perform much better in ensemble prediction. These results are simply implied in a real coupled air–sea model. For the prediction of oceanic variables, using independent perturbations (NLLVs) and adding perturbations to oceanic variables are expected to result in better performance in the ensemble prediction.

Reference

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