Nonlinear Perturbation Method Research Articles

Role of nonlinear transmission term in stability of hand, foot and mouth disease dynamics

Aim: This study focuses on the nonlinear characteristics of two linearly stable equilibrium states within a theoretical hand, foot and mouth disease (HFMD) dynamics model. Methods: The conditional nonlinear optimal perturbation method, the Lyapunov method and the linear singular vector method are employed. Results: By utilizing the conditional nonlinear optimal perturbation method, it is determined that the two linearly stable equilibrium states (namely the free disease and disease HFMD states) also exhibit nonlinear stability. It is found that the stabilities of the free disease and disease HFMD equilibrium states differ in terms of returning to their original states. Conclusion: It is discovered that nonlinearity plays a crucial role in reinstating the original equilibrium states.

An Ensemble CNOP Method Based on a Pre-Screening Mechanism for Targeted Observations in the South China Sea

The uncertainty in the initial condition seriously affects the forecasting skill of numerical models. Targeted observations play an important role in reducing uncertainty in numerical prediction. The conditional nonlinear optimal perturbation (CNOP) method is a useful tool for studying adaptive observation. However, the traditional CNOP method highly relies on the adjoint model, and it is difficult to find the global optimal solution. In this paper, a pre-screening and ensemble CNOP hybrid method called PECNOP is proposed to identify optimal sensitive areas in targeted observations. PECNOP is an adjoint-free method that captures global CNOP with high probability, which can effectively solve the two major problems faced by traditional CNOP methods. We evaluated the performance of PECNOP by building an observation simulation system consisting of an ocean model and data assimilation. One of the assimilation experiments was dedicated to evaluating the stability and effectiveness of PECNOP in extreme events. The results show that, compared with traditional methods, PECNOP can stably capture the global CNOP. Extra observations and assimilation in the optimal sensitive areas identified by PECNOP can effectively improve forecasting by about 20% within 30 days. Therefore, PECNOP has potential to reduce the initial error of numerical models, which is important for improving forecasting.

Modeling and stability analysis of an extended car-following model behavior of the currents vehicle based on delay of response and signal interruption probability

This paper studies the modeling and stability analysis of an extended car-following model based on delay of response and signal interruption probability. First, by linear analysis, we give the stability condition of the model. The result shows that the stability region of the traffic flow will expand with the decrease in the probability p of signal transmission interruption. When the delay of response effect of the driver weakens, the traffic flow will become more and more stable. Second, the mKdV equation is derived based on the nonlinear reduction perturbation method. This result shows that with the decrease of parameter p, the amplitude of the traffic flow density wave decreases, the traffic flow becomes more and more stable, and the traffic congestion is gradually improved. Finally, numerical simulation verifies the correctness of the solution results.

Targeted Observations on Arctic Sea Ice Concentration for Improving Extended‐Range Prediction of Ural Blocking

AbstractThe predictability of certain extreme weather events can exceed the traditional 2 weeks by considering the boundary conditions. Targeted observations in sensitive areas (SA) on Arctic sea ice concentration (SIC) can improve the extended‐range (4 pentads) forecast skills of long‐lasting and strong Ural blocking (UB). The SA are determined based on the SIC optimally growing boundary errors, obtained by the conditional nonlinear optimal perturbation method. The SA are mainly located in the Barents Sea, Greenland Sea, and Okhotsk Sea. The results of observing system simulation experiments for 8 UB cases indicate that the targeted observations can remarkably improve the prediction skills of UB in the fourth pentad. Targeted observations have a positive effect on 75% of 160 experiment members, reduce 35% forecast errors of the fourth pentad mean blocking index, and perform even better when the original forecast errors are greater. Further diagnosis shows that targeted observations contribute to more accurate SIC boundary conditions in the Barents Sea, Greenland Sea, and Okhotsk Sea and reduce temperature errors in the lower and middle troposphere. It further results in well‐described westerly winds in the Ural region and its adjacent regions, corresponding to the more skillful predictions of blocking circulations. The above results supply a theoretical base for the design of Arctic SIC observations and more skillful extended‐range predictions for mid‐latitude extreme weather.

On the Quartic Korteweg–de Vries hierarchy of nonlinear Rossby waves and its dynamics

A long-standing challenge in exploring the dynamics of large-scale atmospheric or oceanic motions is to find more appropriate theoretical models. This paper aims at the mechanisms of nonlinear Rossby waves by a new obtained nonlinear evolutionary equation hierarchy approach. The multiple scales method and weak nonlinear perturbation method are adopted based on the quasi-geostrophic potential vorticity conservation law for large scale atmospheric or oceanic motions. It is the first time that a Quartic Korteweg–de Vries model equation is derived to describe the propagation of nonlinear Rossby waves amplitude, which provides a new acceptable theoretical model for the study of atmospheric blocking phenomena. Qualitative and quantitative analysis for the new Quartic KdV model are finished to disclose the evolutionary mechanisms of nonlinear Rossby waves. The corresponding results declares that the β effect, topographic features and detuning parameter are all important factors on the nonlinear Rossby waves.

Application of the Observation‐Oriented CNOP‐P Sensitivity Analysis Method in Evapotranspiration Simulation and Prediction Over the Tibetan Plateau

AbstractTo improve the ability to accurately simulate and predict evapotranspiration (ET) over the Tibetan Plateau (TP), an observation‐oriented sensitivity analysis (SA) method based on a conditional nonlinear optimal parameter perturbation (CNOP‐P) method is employed. With the observation‐oriented CNOP‐P SA method, a sensitive parameter subset among all parameters is first determined through comparison with observations of ET across the TP. Then, the sensitive parameter combination is optimized to evaluate the ET simulation ability and prediction skill over the TP. A traditional one‐at‐a‐time (OAT) method is also employed, and the results are compared with the numerical results obtained with the observation‐oriented CNOP‐P SA method. The results show that the identified parameter combination using the observation‐oriented CNOP‐P SA method is more sensitive than that using the OAT method for the estimation of ET at 13 sites on the TP. The above results demonstrate that it is important to determine the sensitive parameter combination using the observation‐oriented SA method to improve the ET simulation ability and prediction skill over the TP.

An approach to refining the ground meteorological observation stations for improving PM2.5 forecasts in the Beijing–Tianjin–Hebei region

Abstract. This paper investigates how to refine the ground meteorological observation network to greatly improve the PM2.5 concentration forecasts by identifying sensitive areas for targeted observations that are associated with a total of 48 forecasts in eight heavy haze events during the years of 2016–2018 over the Beijing–Tianjin–Hebei (BTH) region. The conditional nonlinear optimal perturbation (CNOP) method is adopted to determine the sensitive area of the surface meteorological fields for each forecast, and a total of 48 CNOP-type errors are obtained including wind, temperature, and water vapor mixing ratio components. It is found that, although all the sensitive areas tend to locate within and/or around the BTH region, their specific distributions are dependent on the events and the start times of the forecasts. Based on these sensitive areas, the current ground meteorological stations within and around the BTH region are refined to form a cost-effective observation network, which makes the relevant PM2.5 forecasts starting from different initial times for varying events assimilate fewer observations, but overall, it achieve the forecasting skill comparable to and even higher than that obtained by assimilating all ground station observations. This network sheds light on the idea that some of the current ground stations within and around the BTH region are very useless for improving the PM2.5 forecasts in the BTH region and can be greatly scattered to avoid unnecessary work.

Optimal Precursors for Central Pacific El Niño Events in GFDL CM2p1

Abstract Compared to the canonical eastern Pacific El Niño, the understanding and ability to predict the central Pacific (CP)-type event still need further improvement. In this study, the principal component analysis–based particle swarm optimization algorithm (PPSO) is applied in Geophysical Fluid Dynamics Laboratory Climate Model version 2p1 (GFDL CM2p1) to obtain the optimal precursors (OPRs) for CP El Niño events, based on the conditional nonlinear optimal perturbation (CNOP) method. For this, three normal years with neither El Niño nor La Niña events, i.e., three cases, are chosen as the reference states. The obtained OPRs for these cases exhibit a consistent positive sea surface temperature (SST) perturbation distribution in the subtropical North Pacific (20°–40°N, 175°E–140°W), which is further proven to be crucial for the evolution of CP El Niño based on the Northern and Southern Hemisphere significance test results. Mechanically, these positive SST perturbations are enhanced and reach the equatorial Pacific via wind–evaporation–SST (WES) feedback to evolve into a CP El Niño at the end of the year. The nonlinear approach is adopted to investigate the predictability of CP El Niño events and can shed some light on future studies.

Extent of the Impact of Arctic Atmospheric Uncertainty on Extended‐Range Forecasting of Cold Events in East Asia

AbstractThe extended‐range predictability of three simulated extreme cold events in East Asia in the Community Atmosphere Model (version 4) control experiment, whose atmospheric circulation backgrounds are similar to two recent observational cases, is investigated. The results show that they have a predictability of four pentads. Then, we evaluate the extent of the forecast uncertainty in the 4th pentad caused by the initial atmospheric uncertainties in the Arctic, which are large due to sparse instrumental observations. The initial uncertainty leading to the largest forecast uncertainty is obtained by the conditional nonlinear optimal perturbation method, and referred to as the CNOP‐type initial uncertainty. The forecast of the cold surge in the 4th pentad fails after adding the CNOP‐type initial uncertainty at day 0. In comparison, the forecast uncertainties are much weaker when the initial conditions are perturbed by noises, and the weaker influence may be due to the noises' lack of spatial structure. In terms of how the CNOP‐type initial uncertainty develops, a baroclinic structure is seen in the uncertainties on the first 2 days, followed by a propagating Rossby wave feature from the 2nd pentad to the 4th pentad. Meanwhile, synoptic transient eddy feedback also plays an essential role. The results suggest that the CNOP‐type initial uncertainty has potential to identify sensitive areas for targeted observations; plus, it could serve as a member of ensemble initial perturbations, since it indicates the largest uncertainty growth.

Dimension shifting based intelligent algorithm framework to solve conditional nonlinear optimal perturbation

Conditional Nonlinear Optimal Perturbation (CNOP) method is an effective way to study the predictability of oceanic and climatic events. A framework combining the Feature Extraction method and Intelligent Algorithm (FEIA) is frequently used to solve CNOP because it is gradient-free and scalable at high-dimensional scales. However, the fixed latent subspace of FEIA framework makes it challenging to achieve both the quality of solutions and the solving efficiency. To overcome this bottleneck, this paper proposes Dimension Shifting based Intelligent Algorithm (DSIA) framework to solve CNOP in the large-scale model. DSIA framework adopts the dimension shifting strategy, which dynamically shifts search particles in different low-dimensional spaces. To verify the feasibility of DISA framework, we take a Regional Ocean Modeling System (ROMS) model of double-gyre variation as an experimental case. In experiments, we figure out that the selection of feature extraction method and the shifting dimension set are two influential factors for the performance of DSIA framework. Besides, in comparative experiments, DSIA framework yields better objective function values and more valid CNOP than FEIA framework. Moreover, convergence experiments demonstrate DSIA framework can solve CNOP with an appropriate number of function evaluations and has better convergence performance. In conclusion, experimental results prove that DSIA improves both quality of CNOP and solving efficiency.

Error Evolutions and Analyses on Joint Effects of SST and SL via Intermediate Coupled Models and Conditional Nonlinear Optimal Perturbation Method

A seasonal predictability barrier has long been noticed in ENSO forecasting with numerical models. Previous studies explored the impact of seasonal optimal initial perturbation evolutions in sea surface temperature anomalies (SSTA) on ENSO forecasting using the intermediate coupled model (ICM) via the conditional nonlinear optimal perturbation (CNOP) method. In this paper, we investigate the joint effects of SSTA and sea level anomalies (SLA) from the perspective of the optimal growth initial error (OGE). After determining the four seasonal OGEs about SSTA and SLA (i.e., SSTA-OGE, SLA-OGE and Joint-OGE), we first demonstrate the patterns, evolutions and the resulting spring predictability barrier (SPB) of the above OGEs. Then, we analyze the mechanism of OGE evolutions and SPB. Finally, we conduct observing system simulation experiments to determine the best (economic) observation network. Our experimental results indicate that the ENSO evolution error induced by SSTA-OGE and Joint-OGE presents season dependency, but SLA-OGE has no impact on ENSO evolution. Moreover, Joint-OGEs induce error evolutions and the SPB with more significant intensity than SSTA-OGEs and SLA-OGEs. From mechanism analyses, the evolutions of SSTA-OGEs are mainly dominated by Bjerknes feedback. Further, the evolution dynamics of Joint-OGEs primarily contain the continuous heating between the upper ocean combined with Bjerknes feedback and thermal diffusion in response to the discharge process. In addition, comprehensive and economical sensitive areas are identified through Joint-OGE, including the central-eastern equatorial Pacific and the western and north-eastern tropical Pacific boundary, which contribute to the ENSO prediction benefits reaching 58.31% on average.

Applications of CNOP-P Method to Predictability Studies of Terrestrial Ecosystems

In this paper, recent research on terrestrial ecosystem predictability using the conditional nonlinear optimal parameter perturbation (CNOP-P) method is summarized. The main findings include the impacts of uncertainties in climate change on uncertainties in simulated terrestrial ecosystems, the identification of key physical parameters that lead to large uncertainties in terrestrial ecosystem modeling and prediction, and the evaluation of the simulation ability and prediction skill of terrestrial ecosystems by reducing key physical parameter errors. The study areas included the Inner Mongolia region, north–south transect of eastern China, and Qinghai–Tibet Plateau region. The periods of the studies were from 1961 to 1970 for the impacts of uncertainties in climate change on uncertainties in simulated terrestrial ecosystems, and from 1951 to 2000 for the identification of the most sensitive combinations of physical parameters. Climatic Research Unit (CRU) data were employed. The numerical results indicate the important role of nonlinear changes in climate variability due to the occurrences of extreme events characterized by CNOP-P in the abrupt grassland ecosystem equilibrium state and formation of carbon sinks in China. Second, the most sensitive combinations of physical parameters to the uncertainties in simulations and predictions of terrestrial ecosystems identified by the CNOP-P method were more sensitive than those obtained by traditional methods (e.g., one-at-a-time (OAT) and stochastic methods). Furthermore, the improvement extent of the simulation ability and prediction skill of terrestrial ecosystems by reducing the errors of the sensitive physical parameter combinations identified by the CNOP-P method was higher than that by the traditional methods.

Application of the CNOP‐P Ensemble Prediction (CNOP‐PEP) Method in Evapotranspiration Forecasting Over the Tibetan Plateau to Model Parameter Uncertainties

AbstractModel parameter errors are one of the sources of uncertainty when simulating or predicting evapotranspiration (ET) over the Tibetan Plateau (TP). To enhance the ET simulation ability and prediction skill over the TP, the conditional nonlinear optimal parameter perturbation (CNOP‐P) method is used to establish an ensemble prediction method (called the CNOP‐PEP method) to represent uncertainties arising due to the model parameters to implement ensemble prediction experiments. The one‐at‐a‐time (OAT) method and the traditional stochastically perturbed parametrization (SPP) scheme are also employed to implement ensemble prediction experiments for comparison with the CNOP‐PEP method. The numerical results show that all ensemble prediction experiments conducted with the three methods exhibit improved prediction skills compared to the reference ET over the TP. Furthermore, the prediction skill by employing the CNOP‐PEP method is more excellent than those of the OAT and SPP methods when predicting ET over the TP. The physical mechanism analysis shows that the improved evaporation characterization obtained with the ensemble prediction experiments plays a key role in reducing uncertainties when simulating and predicting ET over the TP. The above results indicate that ensemble prediction methods are a helpful tool to improve the simulation ability and prediction skill of the ET over the TP for analyzing model parameters. In addition, the CNOP‐PEP method can supply prime ensemble members to represent uncertainties in model parameters.

Evaluation of the sensitivity on mesoscale eddy associated with the sea surface height anomaly forecasting in the Kuroshio Extension

The sensitivity of the sea surface height anomaly (SSHA) forecasting on the accuracy of mesoscale eddies over the Kuroshio Extension region, which was determined by the conditional non-linear optimal perturbation (CNOP) method using a two-layer quasigeostrophic model, is evaluated by adopting multiply realistic marine datasets through an advanced particle filter assimilation method. It is shown that, if additional observations are preferentially assimilated to the sensitive area of mesoscale eddies identified by the CNOP, where the eddies present a clear high- to low-velocity gradient along the eddy rotation, the forecasting skill of the SSHA can be more significantly improved. It is also demonstrated that the forecasts of the SSHA in the region where the large-scale mean flow possesses much stronger barotropic and/or baroclinic instability tend to exhibit stronger sensitivity to the accuracy of the initial field in the sensitive area of mesoscale eddies. Therefore, more attention should be preferentially paid to the assimilation of the additional observations of the mesoscale eddies for the SSHA forecast in the region with a strong velocity shear of ocean circulation. The present study verifies the sensitivity on mesoscale eddies of SSHA forecasts derived by the two-layer quasigeostrophic model using multiply sets of realistic oceanic data, especially including observation and reanalysis data, which further additionally demonstrates the importance of targeted observations of mesoscale eddies to the SSHA forecast in the regions of strong velocity shear of ocean circulation and provides a more credible scientific basis for the field campaign of the targeted observations for mesoscale eddies associated with the SSHA forecasting.

A Biological‐Parameter‐Optimized Modeling Study of Physical Drivers Controlling Seasonal Chlorophyll Blooms off the Southern Coast of Java Island

AbstractA physical‐biogeochemical model consists of many key parameters; theoretically, 10,000 or more numerical experiments are required to obtain a set of optimal biological parameters. The corresponding computing resources are too expensive when running a three‐dimensional and high‐resolution physical‐biogeochemical model. In this study, we improve the performance of a three‐dimensional physical‐biogeochemical model via the conditional nonlinear optimal perturbation method for southern Java Island, where chlorophyll blooms occur seasonally. Based on the improved model results, we calculated the mixed‐layer nutrient budget, thereby characterizing how each physical process transports nutrients into the surface oligotrophic layer and then stimulates the chlorophyll blooms during the southeast monsoon season. Nutrients that upwell along the submarine slope are found in narrow regions along the coast and spread into most of the study area by local wind‐induced offshore transport. The contribution of nonlinearity (advection) is essential to modulating the upslope movement of nutrients off the coast. If local winds are neglected in the model, onshore transport induced by the pressure gradient force in the zonal direction (alongshore) and subsequent downwelling would be dominant, and chlorophyll blooms would diminish in most regions. The role of the South Java Coastal Current varies under the joint influence of remote and local wind forcings. The Indonesian Throughflow from the Lombok Strait also contributes to surface phytoplankton growth along its path off the south of Java Island.

A novel nonlinear FE perturbation method and its application to stacking sequence optimization for snap-through response of cylindrical shell panel

Cylindrical shell panels are commonly used for the lightweight design of thin-walled structures in the aerospace engineering. In this work, the snap-through buckling of composite cylindrical shell panels subjected to lateral loads is calculated and designed. We present a novel method to extract the essential features of the snap-through response using a fast path-following scheme, and subsequently apply them as inputs to the stacking sequence optimization. In the path-following scheme, the full snap-through response can be achieved efficiently based on the finite element perturbation method, meanwhile its numerical accuracy is well guaranteed by the necessary corrections on residual forces. The essential features of the snap-through response, i.e. the load-limit point, critical deflection and jump deflection, are selected to be either the objective or the constraint in the stacking sequence optimization. The stacking sequence optimization using the genetic algorithm can be realized considering the severe geometrical nonlinearities of mechanics responses, benefiting from the high efficiency and robustness of a single function call. Composite cylindrical shell panels with various geometry/lamination configurations and load/displacement boundary conditions, are considered in the stacking sequence optimizations, to validate the good performance of the proposed method.

Optimal Precursors Identification for North Atlantic Oscillation Using the Parallel Intelligence Algorithm

The North Atlantic Oscillation (NAO) with abnormal sea level pressure (SLP) differences influence the westerly wind strength and storm tracks in the North Atlantic, which further affects the winter climate in the northern hemisphere. The predictability of NAO has become an important area of climate research in recent years. The identification of optimal precursors (OPRs) would help to investigate the dynamics of atmospheric and oceanic motions, as well as the nonlinear characteristic. The conditional nonlinear optimal perturbation (CNOP) method has been widely used in research on the OPR of the climatic event to explore which kind of initial perturbation is most likely to trigger climate events. However, the previous works on NAO’s OPR are based on a simple ideal model, which cannot describe the evolution process of NAO. Moreover, the commonly used algorithms that rely on the adjoint model are not suitable for the large complicated numerical models that do not have corresponding adjoint models, like Community Earth System Model (CESM). To break through the limitation, this paper proposes the parallel principal component analysis (PCA)-based particle swarm optimization (PSO) and genetic algorithm (GA) hybrid algorithm (PGAPSO) algorithm to identify the OPR of NAO using CESM. For different initial conditions, OPRs identified by the proposed method can always make the basic state develop into NAO events with corresponding phases. Compared with other adjoint-free approaches, the proposed method has relatively better fitness values and more mature patterns. The results also reveal that the proposed method can avoid falling into a local optimum and has strong robustness. In terms of performance, the proposed method searches in feature space with a much lower number of dimensions, thus improving efficiency. In addition, the proposed method is accelerated with multiple parallel frameworks to enhance performance, and it achieves the speedup ratio of 40.2.

Optimally growing initial errors of El Ni\xf1o events in the CESM

The optimally growing initial errors (OGEs) of El Niño events are found in the Community Earth System Model (CESM) by the conditional nonlinear optimal perturbation (CNOP) method. Based on the characteristics of low-dimensional attractors for ENSO (El Niño Southern Oscillation) systems, we apply singular vector decomposition (SVD) to reduce the dimensions of optimization problems and calculate the CNOP in a truncated phase space by the differential evolution (DE) algorithm. In the CESM, we obtain three types of OGEs of El Niño events with different intensities and diversities and call them type-1, type-2 and type-3 initial errors. Among them, the type-1 initial error is characterized by negative SSTA errors in the equatorial Pacific accompanied by a negative west–east slope of subsurface temperature from the subsurface to the surface in the equatorial central-eastern Pacific. The type-2 initial error is similar to the type-1 initial error but with the opposite sign. The type-3 initial error behaves as a basin-wide dipolar pattern of tropical sea temperature errors from the sea surface to the subsurface, with positive errors in the upper layers of the equatorial eastern Pacific and negative errors in the lower layers of the equatorial western Pacific. For the type-1 (type-2) initial error, the negative (positive) temperature errors in the eastern equatorial Pacific develop locally into a mature La Niña (El Niño)-like mode. For the type-3 initial error, the negative errors in the lower layers of the western equatorial Pacific propagate eastward with Kelvin waves and are intensified in the eastern equatorial Pacific. Although the type-1 and type-3 initial errors have different spatial patterns and dynamic growing mechanisms, both cause El Niño events to be underpredicted as neutral states or La Niña events. However, the type-2 initial error makes a moderate El Niño event to be predicted as an extremely strong event.

First- and second-order crossover between quantum and classical regimes for the escape rate of a nanomagnet manipulated by a Josephson current

Employing the instanton technique and the nonlinear perturbation method, we study quantum-classical escape rate crossover for several forms of magnetic anisotropy in a nanomagnet sandwiched between two superconductors called Josephson φ0 junction. It is shown that the phase boundary separating the first- and second-order crossover is greatly influenced by the bias current as well as the anisotropy constant and the external magnetic field. These features are expected to be observable with existing experimental techniques.

A lattice method for seismic wave propagation in nonlinear viscoelastic media

SUMMARYNonlinear wave propagation is an important consideration in several geophysical problems as heterogeneities within the subsurface give rise to nonlinear stress–strain relationships in rocks. To address these problems, where the inclusion of subsurface complexity is accounted for, numerical solutions are required. Here, we outline a discrete particle or lattice numerical method that can simulate dynamic and static deformation in an isotropic viscoelastic nonlinear medium where the nonlinearity includes the fourth-order elastic tensor coefficients. The relationship between the local interaction constants in the lattice method and the macroscopic nonlinear coefficients is derived and the relationships presented. The theoretical dispersion properties of the system is determined from a nonlinear perturbation method and are validated against some numerical results. There is an excellent agreement between the theory and numerical results and the method reproduces the common features of nonlinear wave propagation, for example, harmonics, waveform distortion and spectra shifts. As expected the system is shown to be dependent on the wave amplitudes and an example is presented which shows the complexity that can arise in both dynamic and static deformation in a complex model where nonlinearity is included.