Nonlinear Local Lyapunov Exponent Research Articles

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

Extreme Events Prediction from Nonlocal Partial Information in a Spatiotemporally Chaotic Microcavity Laser.

The forecasting of high-dimensional, spatiotemporal nonlinear systems has made tremendous progress with the advent of model-free machine learning techniques. However, in real systems it is not always possible to have all the information needed; only partial information is available for learning and forecasting. This can be due to insufficient temporal or spatial samplings, to inaccessible variables, or to noisy training data. Here, we show that it is nevertheless possible to forecast extreme event occurrences in incomplete experimental recordings from a spatiotemporally chaotic microcavity laser using reservoir computing. Selecting regions of maximum transfer entropy, we show that it is possible to get higher forecasting accuracy using nonlocal data vs local data, thus allowing greater warning times of at least twice the time horizon predicted from the nonlinear local Lyapunov exponent.

The backward nonlinear local Lyapunov exponent and its application to quantifying the local predictability of extreme high-temperature events

The backward nonlinear local Lyapunov exponent and its application to quantifying the local predictability of extreme high-temperature events

A New Technique to Quantify the Local Predictability of Extreme Events: The Backward Nonlinear Local Lyapunov Exponent Method

Extreme weather events have a large impact on society, but are challenging to forecast accurately. In this study, we carried out a theoretical investigation of the local predictability of extreme weather events using the Lorenz model. We introduce a new method using the backward nonlinear local Lyapunov exponent to quantitatively estimate the local predictability limits of extreme events. The local predictability limits of extreme events on an individual orbit of a dynamical trajectory are broadly the same, whereas this is not the case if they are on different orbits. The specific structure of the Lorenz attractor is responsible for this phenomenon. Our results show that the local predictability limits of extreme events do not decrease or increase monotonically as the events increase in magnitude. This indicates that the magnitude of extreme events is not the only factor that affects the local predictability. The dynamical flow, initial error size, and structure of an attractor may also affect the local predictability. We also quantitatively compared the local predictability of extreme warm and cold events. This showed that the local predictability limits of extreme warm events are higher than extreme cold events at the same probability. A statistical analysis (i.e., the minimum, first quartile, median, third quartile, and maximum) also suggests that the extreme warm events have higher local predictability limits. In general, extreme warm events are more predictable than extreme cold events.

Investigating decadal variations of the seasonal predictability limit of sea surface temperature in the tropical Pacific

El Niño and the Southern Oscillation (ENSO) have a worldwide impact on seasonal to yearly climate. However, there are decadal variations in the seasonal prediction skill of ENSO in dynamical and statistical models; in particular, ENSO prediction skill has declined since 2000. The shortcomings of models mean that it is very important to study ENSO seasonal predictability and its decadal variation using observational/reanalysis data. Here we quantitatively estimate the seasonal predictability limit (PL) of ENSO from 1900 to 2015 using Nonlinear local Lyapunov exponent (NLLE) theory with an observational/reanalysis dataset and explore its decadal variations. The mean PL of sea surface temperature (SST) is high in the central/eastern tropical Pacific and low in the western tropical Pacific, reaching 12–15 and 7–8 months, respectively. The PL in the tropical Pacific varies on a decadal timescale, with an interdecadal standard deviation of up to 2 months in the central tropical Pacific that has similar spatial structure to the mean PL. Taking the PL of SST in the Niño 3.4 region as representative of the PL in the central/eastern tropical Pacific, there are clearly higher values in the 1900s, mid-1930s, mid-1960s, and mid-1990s, and lower values in the 1920s, mid-1940s, and mid-2010s. Meanwhile, the PL of SST in the Niño 6 region—whose average value is 7 months—is in good agreement with the PL of most regions in the western tropical Pacific, with higher values in the 1910s, 1940s, and 1980s and lower values in the 1930s, 1950s, and mid-1990s. In the framework of NLLE theory, the PL is determined by the error growth rate (representing the dissipation rate of the predictable signal) and the saturation value of relative error (representing predictable signal intensity). We reveal that the spatial structure of the mean PL in the tropical Pacific is determined mainly by the error growth rate. The decadal variability of PL is affected more by the variation of the saturation value of relative error in the equatorial Pacific, whereas the error growth rate cannot be ignored in the PL of some regions. As an important source of predictability in ENSO dynamics, the relationship between warm water volume and SST in the Niño 3.4 region has a critical role in the decadal variability of PL in the tropical Pacific through the error growth rate and saturation value of relative error. This strong relationship reduces the error growth rate in the initial period and increases the saturated relative error, contributing to the high PL.

Predictability Limit of Monsoon Intraseasonal Precipitation: An Implication of Central Indian Ocean Mode

The prediction of monsoonal precipitation during Indian summer monsoon (ISM) remains difficult. Due to the high correlation between the Central Indian Ocean (CIO) mode index and the ISM precipitation variability, the predictability limit of the CIO mode index is investigated by the non-linear local Lyapunov exponent (NLLE) method in observations. Results show that the predictability limit of the CIO mode index can reach 38 days during boreal summer (from June to September), which is close to the upper predictability limit of intraseasonal precipitation (up to 40 days), and higher than the predictability limits of dynamical monsoon indices (under 3 weeks) and boreal summer intraseasonal oscillation (BSISO) indices (around 30 days). Such high predictability limit of the CIO mode index is mainly attributable to the long predictability limits from the intraseasonal sea surface temperature (SST) and intraseasonal zonal wind, which are the components of the CIO mode. As a result, the CIO mode is expected to extend the predictability of monsoonal precipitation, and benefits to improve the prediction skills of the ISM.

Application of Backward Nonlinear Local Lyapunov Exponent Method to Assessing the Relative Impacts of Initial Condition and Model Errors on Local Backward Predictability

Application of Backward Nonlinear Local Lyapunov Exponent Method to Assessing the Relative Impacts of Initial Condition and Model Errors on Local Backward Predictability

The Predictability Limit of Ocean Mesoscale Eddy Tracks in the Kuroshio Extension Region

In this study, the nonlinear local Lyapunov exponent and nonlinear error growth dynamics are employed to estimate the predictability limit of oceanic mesoscale eddy (OME) tracks quantitatively using three datasets. The results show that the mean predictability limit of OME tracks is about 53 days for cyclonic eddy (CE) and 52 days for anticyclonic eddy (AE) in the Kuroshio Extension (KE). The predictability limit varies spatially. The predictability limit of OME tracks is higher for the eastern region (about 62.5 days) than that for the western part (about 46 days). The CEs (AEs) predictability limit is relatively high in the southern (northern) region. Additionally, the lifetime, amplitude, and radius of OME are closely related to the predictability limit. The long-lived, large-amplitude, and large-sized OMEs tend to be more predictable. The eastern region often generates long-lived and large-size OMEs, thereby obtaining a higher predictability limit of OME tracks. Furthermore, the relationship between the predictability limit and the smoothness of the OME tracks was investigated using a metric to describe the track’s complexation. Usually, OMEs with high predictability limit values often show extender and smoother trajectories. The effects of the surface ocean circulations and the surface winds are also investigated. The strong and energetic currents lead to a short limitation in the west region.

A multi-model study of atmosphere predictability in coupled ocean\u2013atmosphere systems

Of great importance for guiding numerical weather and climate predictions, understanding predictability of the atmosphere in the ocean − atmosphere coupled system is the first and critical step to understand predictability of the Earth system. However, previous predictability studies based on prefect model assumption usually depend on a certain model. Here we apply the predictability study with the Nonlinear Local Lyapunov Exponent and Attractor Radius to the products of multiple re-analyses and forecast models in several operational centers to realize general predictability of the atmosphere in the Earth system. We first investigated the predictability characteristics of the atmosphere in NCEP, ECMWF and UKMO coupled systems and some of their uncoupled counterparts and other uncoupled systems. Although the ECMWF Integrated Forecast System shows higher skills in geopotential height over the tropics, there is no certain model providing the most precise forecast for all variables on all levels and the multi-model ensemble not always outperforms a single model. Improved low-frequency signals from the air − sea and stratosphere − troposphere interactions that extend predictability of the atmosphere in coupled system suggests the significance of air − sea coupling and stratosphere simulation in practical forecast development, although uncertainties exist in the model representation for physical processes in air − sea interactions and upper troposphere. These inspire further exploration on predictability of ocean and stratosphere as well as sea − ice and land processes to advance our understanding of interactions of Earth system components, thus enhancing weather − climate prediction skills.

Decreasing predictability as a precursor indicator for abrupt climate change

Abrupt climate change has an important impact on sustainable economic and social development, as well as ecosystem. However, it is very difficult to predict abrupt climate changes because the climate system is a complex and nonlinear system. In the present paper, the nonlinear local Lyapunov exponent (NLLE) is proposed as a new early warning signal for an abrupt climate change. The performance of NLLE as an early warning signal is first verified by those simulated abrupt changes based on four folding models. That is, NLLE in all experiments showed an almost monotonous increasing trend as a dynamic system approached its tipping point. For a well-studied abrupt climate change in North Pacific in 1976/1977, it is also found that NLLE shows an almost monotonous increasing trend since 1970 which give up to 6 years warning before the abrupt climate change. The limit of the predictability for a nonlinear dynamic system can be quantitatively estimated by NLLE, and lager NLLE of the system means less predictability. Therefore, the decreasing predictability may be an effective precursor indicator for abrupt climate change.

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

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.

Quantitative study of the relative effects of initial condition and model uncertainties on local predictability in a nonlinear dynamical system

Abstract The relative effects of initial condition and model uncertainties on local predictability are important issues in the atmospheric sciences. This study quantitatively compared the relative effects of these two types of uncertainty on local predictability using the Lorenz model. Local predictability limits were quantitatively estimated using the nonlinear local Lyapunov exponent (NLLE) method. Results show that the relative effects of initial conditions and model uncertainties on local predictability vary with the state. In addition, inverse spatial distributions of local predictability limits are induced by the two types of uncertainty. In the regime transition region, the local predictability limits of modeled states are more sensitive to initial condition uncertainty than to model uncertainty, resulting in lower local predictability limits being induced by initial condition uncertainties. Local predictability limits induced by initial condition uncertainties are 4 time units shorter than those induced by model uncertainties. In the “butterfly wing” regions, the local predictability limits of modeled states are more sensitive to model uncertainty than to initial condition uncertainty, resulting in lower local predictability limits due to model uncertainty. Local predictability limits induced by initial condition uncertainty are larger (0 to 4 time units) than those induced by model uncertainty. These differences in the regions that are sensitive to the two types of uncertainty mean that strategic reductions of uncertainty in sensitive areas may effectively improve forecast skill.

The predictability limit of the amplitude and phase of the Madden‐Julian oscillation

AbstractThe Madden–Julian Oscillation (MJO) is characterized by slowly eastward‐propagating precipitation and circulation anomalies with time scales of about 30–80 days. Both the phase and amplitude of the MJO fluctuate with time as it propagates eastward. Despite recent progress in understanding the predictability limit of the MJO as a whole, little is known of the difference in the predictability limits of its amplitude and phase. This paper investigates these differences using the nonlinear local Lyapunov exponent approach, which provides an estimate of atmospheric predictability based on observational data. The predictability limit of the phase of the MJO is determined as ~32 days, which is higher than that of its amplitude (about 16 days). In state‐of‐the‐art operational forecast models, the phase of the MJO is also found to have a much better forecast skill than does its amplitude. The relatively low limit of the predictability of the amplitude will pose a challenge to MJO prediction.

Determination of the Backward Predictability Limit and Its Relationship with the Forward Predictability Limit

In this work, two types of predictability are proposed—forward and backward predictability—and then applied in the nonlinear local Lyapunov exponent approach to the Lorenz63 and Lorenz96 models to quantitatively estimate the local forward and backward predictability limits of states in phase space. The forward predictability mainly focuses on the forward evolution of initial errors superposed on the initial state over time, while the backward predictability is mainly concerned with when the given state can be predicted before this state happens. From the results, there is a negative correlation between the local forward and backward predictability limits. That is, the forward predictability limits are higher when the backward predictability limits are lower, and vice versa. We also find that the sum of forward and backward predictability limits of each state tends to fluctuate around the average value of sums of the forward and backward predictability limits of sufficient states. Furthermore, the average value is constant when the states are sufficient. For different chaotic systems, the average value is dependent on the chaotic systems and more complex chaotic systems get a lower average value. For a single chaotic system, the average value depends on the magnitude of initial perturbations. The average values decrease as the magnitudes of initial perturbations increase.

Estimates of Decadal Climate Predictability From an Interactive Ensemble Model

AbstractDecadal climate predictability has received considerable scientific interest in recent years, yet the limits and mechanisms for decadal predictability are currently not well known. It is widely accepted that noise due to internal atmospheric dynamics at the air‐sea interface influences predictability. The purpose of this paper is to use the interactive ensemble (IE) coupling strategy to quantify how internal atmospheric noise at the air‐sea interface impacts decadal predictability. The IE technique can significantly reduce internal atmospheric noise and has proven useful in assessing seasonal‐to‐interannual variability and predictability. Here we focus on decadal timescales and apply the nonlinear local Lyapunov exponent method to the Community Climate System Model comparing control simulations with IE simulations. This is the first time the nonlinear local Lyapunov exponent has been applied to the state‐of‐the‐art coupled models. The global patterns of decadal predictability are discussed from the perspective of internal atmospheric noise and ocean dynamics.

Estimating the Predictability Limit of Tropical Cyclone Tracks over the Western North Pacific Using Observational Data

In this study, the nonlinear local Lyapunov exponent (NLLE) approach was used to quantitatively determine the predictability limit of tropical cyclone (TC) tracks based on observed TC track data obtained from the Joint Typhoon Warning Center. The results show that the predictability limit of all TC tracks over the whole western North Pacific (WNP) basin is about 102 h, and the average lifetime of all TC tracks is about 174 h. The predictability limits of the TC tracks for short-, medium-, and long-lived TCs are approximately 72 h, 120 h, and 132 h, respectively. The predictability limit of the TC tracks depends on the TC genesis location, lifetime, and intensity, and further analysis indicated that these three metrics are closely related. The more intense and longer-lived TCs tend to be generated on the eastern side of the WNP (EWNP), whereas the weaker and shorter-lived TCs tend to form in the west of the WNP (WWNP) and the South China Sea (SCS). The relatively stronger and longer-lived TCs, which are generated mainly in the EWNP, have a longer travel time before they curve northeastwards and hence tend to be more predictable than the relatively weaker and shorter-lived TCs that form in the WWNP region and SCS. Furthermore, the results show that the predictability limit of the TC tracks obtained from the best-track data may be underestimated due to the relatively short observational records currently available. Further work is needed, employing a numerical model to assess the predictability of TC tracks.

Asymmetry of the Predictability Limit of the Warm ENSO Phase

AbstractA nonlinear local Lyapunov exponent method based on monthly sea surface temperature data is employed to explore the predictability limit of warm El Niño–Southern Oscillation (ENSO) events. Results using observational data show an asymmetry of the predictability limit between the developing and decaying stages of the warm ENSO phase. To wit, predictability of the developing stage of warm ENSO events is found to approach a limit of 10 months, less than that of the mature and decaying stages. This asymmetrical predictability limit is also found in a long climate model simulation and may explain the asymmetry in operational forecast skill of warm ENSO events. Through exploring the error growth rate as represented by nonlinear local Lyapunov exponent and the instantaneous error growth rate, it is shown that error growth, especially during the first 8‐month lead forecasts, is the primary contributor to the asymmetry of the predictability limit of warm ENSO events.

Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system

For an n-dimensional chaotic system, we extend the definition of the nonlinear local Lyapunov exponent (NLLE) from one- to n-dimensional spectra, and present a method for computing the NLLE spectrum. The method is tested on three chaotic systems with different complexity. The results indicate that the NLLE spectrum realistically characterizes the growth rates of initial error vectors along different directions from the linear to nonlinear phases of error growth. This represents an improvement over the traditional Lyapunov exponent spectrum, which only characterizes the error growth rates during the linear phase of error growth. In addition, because the NLLE spectrum can effectively separate the slowly and rapidly growing perturbations, it is shown to be more suitable for estimating the predictability of chaotic systems, as compared to the traditional Lyapunov exponent spectrum.

Quantifying local predictability of the Lorenz system using the nonlinear local Lyapunov exponent

AbstractThe nonlinear local Lyapunov exponent (NLLE) can be used as a quantification of the local predictability limit of chaotic systems. In this study, the phase-spatial structure of the local predictability limit over the Lorenz-63 system is investigated. It is found that the inner and outer rims of each regime of the attractor have a high probability of a longer than average local predictability limit, while the center part is the opposite. However, the distribution of the local predictability limit is nonuniformly organized, with adjacent points sometimes showing quite distinct error growth. The source of local predictability is linked to the local dynamics, which is related to the region in the phase space and the duration on the current regime.

Baseline predictability of daily east Asian summer monsoon circulation indices

The nonlinear local Lyapunov exponent (NLLE) method is adopted to quantitatively determine the predictability limit of East Asian summer monsoon (EASM) intensity indices on a synoptic timescale. The predictability limit of EASM indices varies widely according to the definitions of indices. EASM indices defined by zonal shear have a limit of around 7 days, which is higher than the predictability limit of EASM indices defined by sea level pressure (SLP) difference and meridional wind shear (about 5 days). The initial error of EASM indices defined by SLP difference and meridional wind shear shows a faster growth than indices defined by zonal wind shear. Furthermore, the indices defined by zonal wind shear appear to fluctuate at lower frequencies, whereas the indices defined by SLP difference and meridional wind shear generally fluctuate at higher frequencies. This result may explain why the daily variability of the EASM indices defined by zonal wind shear tends be more predictable than those defined by SLP difference and meridional wind shear. Analysis of the temporal correlation coefficient (TCC) skill for EASM indices obtained from observations and from NCEP’s Global Ensemble Forecasting System (GEFS) historical weather forecast dataset shows that GEFS has a higher forecast skill for the EASM indices defined by zonal wind shear than for indices defined by SLP difference and meridional wind shear. The predictability limit estimated by the NLLE method is shorter than that in GEFS. In addition, the June-September average TCC skill for different daily EASM indices shows significant interannual variations from 1985 to 2015 in GEFS. However, the TCC for different types of EASM indices does not show coherent interannual fluctuations.