Conditional Nonlinear Optimal 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.

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.

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.

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.

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 New Scheme for Capturing Global Conditional Nonlinear Optimal Perturbation

Conditional nonlinear optimal perturbation (CNOP) represents the initial perturbation that satisfies a certain physical constraint condition, and leads to a maximum prediction error at the moment of prediction. The CNOP method is a useful tool in studying atmosphere and ocean predictability problems. Generally, the optimization algorithm based on the gradient of the cost function to compute CNOP requires an initial guess. The traditional scheme randomly chooses the initial guess of CNOP within the constraint range and therefore this scheme is called RIG-CNOP. However, the RIG-CNOP scheme reduces the probability of capturing the global CNOP in many cases, such as the prediction model is strongly nonlinear or long-term prediction is performed, or multiple extreme values existed in the cost function. Considering the limitations of the RIG-CNOP scheme, we propose a new initial guess selection scheme. In this scheme, we first pre-analyze a series of random initial guesses, and then, an optimal initial guess is selected. The above process replaces the initial guess selection scheme in the traditional scheme, which is called PAIG-CNOP. Numerical experiments are conducted utilizing the Lorenz-63 model. Also, to compare the performance of the PAIG-CNOP method with the RIG-CNOP method in capturing global CNOP, the CNOP and the maximum cost function value (MCFV) obtained by the filtering method (FM) are used as benchmarks (this value is called FMMCFV in brief). The experimental results show that even the prediction model is strongly nonlinear or the prediction time is long, or the cost function has multiple extreme values, the PAIG-CNOP method can capture the global CNOP with a high probability. The results show that the PAIG-CNOP method has a higher probability of capturing the global CNOP than the RIG-CNOP method. In addition, we use an ensemble-based technique in the computation of gradients, thus avoiding the use of adjoint techniques in the maximization process. Due to the attractive features of the new method, the PAIG-CNOP method is an efficient and useful method for solving CNOP, it can be more easily applied to obtain the global CNOP of operational prediction models.

Targeted Observations for Improving Prediction of the NAO Onset

Based on the viewpoint that the North Atlantic Oscillation (NAO) has an intrinsic timescale of approximate two weeks and can be treated as an initial value problem, targeted observations for improving the prediction of the onset of NAO events are investigated by using the conditional nonlinear optimal perturbation (CNOP) method with a quasi-geostrophic model. The results show that flow-dependent sensitive areas for the prediction of NAO onset are mainly located over North Atlantic and its upstream regions. Targeted observations over the main sensitive areas could improve NAO onset prediction in most cases (approximately 75%) due to reduced errors in anomalous eddy vorticity forcing (EVF) projection in the typical NAO mode. Moreover, a flow-independent sensitive area is determined based on the winter climatological flow, which is located over North America and its adjacent ocean. The NAO onset prediction can also be improved by targeted observations over the flow-independent sensitive area, but the skill improvement is somewhat lower than that derived from observations over the flow-dependent sensitive area. The above results indicate that targeted observations over sensitive areas identified by the CNOP method can help to improve the onset prediction of NAO events.

Nonlinear Character Analysis for Bistability in virus–immune Dynamics

Structured abstract Aim: The nonlinear characters of two linearly stable equilibrium states (virus and immune) for a theoretical virus-immune model are analyzed. Methods: Conditional nonlinear optimal perturbation (CNOP), Lyapunov method and linear singular vector method. Results & conclusion: Two linearly stable equilibrium states (immune-free and immune) with linear methods are nonlinearly unstable using the CNOP method. When the CNOP-type of initial perturbation is used in the model, the immune-free (immune) equilibrium state will be made into the immune (immune-free) equilibrium state. Through computing the variations of nonlinear terms of the model, the nonlinear effect of immune proliferation plays an important role in abrupt changes of the immune-free equilibrium state compared with the linear term. For the immune equilibrium state, the nonlinear effect of viral replication is also an important factor.

An Adjoint-Free CNOP–4DVar Hybrid Method for Identifying Sensitive Areas Targeted Observations: Method Formulation and Preliminary Evaluation

This paper proposes a hybrid method, called CNOP–4DVar, for the identification of sensitive areas in targeted observations, which takes the advantages of both the conditional nonlinear optimal perturbation (CNOP) and four-dimensional variational assimilation (4DVar) methods. The proposed CNOP–4DVar method is capable of capturing the most sensitive initial perturbation (IP), which causes the greatest perturbation growth at the time of verification; it can also identify sensitive areas by evaluating their assimilation effects for eliminating the most sensitive IP. To alleviate the dependence of the CNOP–4DVar method on the adjoint model, which is inherited from the adjoint-based approach, we utilized two adjoint-free methods, NLS-CNOP and NLS-4DVar, to solve the CNOP and 4DVar sub-problems, respectively. A comprehensive performance evaluation for the proposed CNOP–4DVar method and its comparison with the CNOP and CNOP–ensemble transform Kalman filter (ETKF) methods based on 10 000 observing system simulation experiments on the shallow-water equation model are also provided. The experimental results show that the proposed CNOP–4DVar method performs better than the CNOP–ETKF method and substantially better than the CNOP method.

A useful approach to sensitivity and predictability studies in geophysical fluid dynamics: conditional non-linear optimal perturbation.

In atmospheric and oceanic studies, it is important to investigate the uncertainty of model solutions. The conditional non-linear optimal perturbation (CNOP) method is useful for addressing the uncertainty. This paper reviews the development of the CNOP method and its computational aspects in recent years. Specifically, the CNOP method was first proposed to investigate the effects of the optimal initial perturbation on atmosphere and ocean model results. Then, it was extended to explore the influences of the optimal parameter perturbation, model tendency perturbation and boundary condition perturbation. To obtain solutions to these optimal perturbations, four kinds of optimization approaches were developed: the adjoint-based method, the adjoint-free method, the intelligent optimization method and the unconstrained optimization method. We illustrate the calculation process of each method and its advantages and disadvantages. Then, taking the Zebiak–Cane model as an example, we compare the CNOPs related to initial conditions (CNOP-Is) calculated by the above four methods. It was found that the dominant structures of the CNOP-Is for different methods are similar, although some differences in details exist. Finally, we discuss the necessity and possible direction for designing a more effective optimization approach related to the CNOP in the future.

Applications of nonlinear optimization approach to atmospheric and oceanic sciences

This paper mainly introduces recent applications of nonlinear optimization approach to atmospheric and oceanic sciences. Emphasis is placed on the theoretical framework of the conditional nonlinear optimal perturbation (CNOP) method that is based on nonlinear optimization, and the works aiming to make it comprehensive. The application progresses of the CNOP method in atmospheric and oceanic sciences are briefly presented, including the applications for ensemble forecast, predictability of some high-impact ocean-atmospheric environmental events, recognition of model parameter sensitivities, assessments of model tendency error and boundary condition error. In addition, we also discuss the difficulties and challenges for the application of the CNOP approach and suggest directions for future development.

Key synoptic-scale features influencing the high-impact heavy rainfall in Beijing, China, on 21 July 2012

This work examined quantitatively the key synoptic features influencing the high-impact heavy rainfall event in Beijing, China, on 21 July 2012 using both correlation analysis based on global ensemble forecasts (from TIGGE) and a method previously used for observation targeting. The global models were able to capture the domain-averaged rainfall of >100 mm but underestimated rainfall beyond 200 mm with an apparent time lag. In this particular case, the ensemble forecasts of the National Centres for Environmental Prediction (NCEP) had apparently better performance than those of the European Centre for Medium-Range Weather Forecasts (ECMWF) and the China Meteorological Administration (CMA), likely because of their high accuracies in capturing the key synoptic features influencing the rainfall event. Linear correlation coefficients between the 24-h domain-averaged precipitation in Beijing and various variables during the rainfall were calculated based on the grand ensemble forecasts from ECMWF, NCEP and CMA. The results showed that the distribution of the precipitation was associated with the strength and the location of a mid-level trough in the westerly flow and the associated low-level low. The dominant system was the low-level low, and a stronger low with a location closer to the Beijing area was associated with heavier rainfall, likely caused by stronger low-level lifting. These relationships can be clearly seen by comparing a good member with a bad member of the grand ensemble. The importance of the trough in the westerly flow and the low-level low was further confirmed by the sensitive area identified through sensitivity analyses with conditional nonlinear optimal perturbation method.

Identifying sensitive areas of adaptive observations for prediction of the Kuroshio large meander using a shallow-water model

Sensitive areas for prediction of the Kuroshio large meander using a 1.5-layer, shallow-water ocean model were investigated using the conditional nonlinear optimal perturbation (CNOP) and first singular vector (FSV) methods. A series of sensitivity experiments were designed to test the sensitivity of sensitive areas within the numerical model. The following results were obtained: (1) the effect of initial CNOP and FSV patterns in their sensitive areas is greater than that of the same patterns in randomly selected areas, with the effect of the initial CNOP patterns in CNOP sensitive areas being the greatest; (2) both CNOP-and FSV-type initial errors grow more quickly than random errors; (3) the effect of random errors superimposed on the sensitive areas is greater than that of random errors introduced into randomly selected areas, and initial errors in the CNOP sensitive areas have greater effects on final forecasts. These results reveal that the sensitive areas determined using the CNOP are more sensitive than those of FSV and other randomly selected areas. In addition, ideal hindcasting experiments were conducted to examine the validity of the sensitive areas. The results indicate that reduction (or elimination) of CNOP-type errors in CNOP sensitive areas at the initial time has a greater forecast benefit than the reduction (or elimination) of FSV-type errors in FSV sensitive areas. These results suggest that the CNOP method is suitable for determining sensitive areas in the prediction of the Kuroshio large-meander path.

Three-Dimensional Structure of Optimal Nonlinear Excitation for Decadal Variability of the Thermohaline Circulation

AbstractThe decadal variability of the North Atlantic thermohaline circulation (THC) is investigated within a three-dimensional ocean circulation model using the conditional nonlinear optimal perturbation method. The results show that the optimal initial perturbations of temperature and salinity exciting the strongest decadal THC variations have similar structures: the perturbations are mainly in the northwestern basin at a depth ranging from −1500 to −3000 m. These temperature and salinity perturbations act as the optimal precursors for future modifications of the THC, highlighting the importance of observations in the northwestern basin to monitor the variations of temperature and salinity at depth. The decadal THC variation in the nonlinear model initialized by the optimal salinity perturbations is much stronger than that caused by the optimal temperature perturbations, indicating that salinity variations might play a relatively important role in exciting the decadal THC variability. Moreover, the deca...

Exploring the phase-strength asymmetry of the North Atlantic Oscillation using conditional nonlinear optimal perturbation

Negative-phase North Atlantic Oscillation (NAO) events are generally stronger than positive-phase ones, i.e., there is a phase-strength asymmetry of the NAO. In this work, we explore this asymmetry of the NAO using the conditional nonlinear optimal perturbation (CNOP) method with a three-level global quasi-geostrophic spectral model. It is shown that, with winter climatological flow forcing, the CNOP method identifies the perturbations triggering the strongest NAO event under a given initial constraint. Meanwhile, the phase-strength asymmetry characteristics of the NAO can be revealed. By comparing with linear results, we find that the process of perturbation self-interaction promotes the onset of negative NAO events, which is much stronger than during positive NAO onset. Results are obtained separately using the climatological and zonal-mean flows in boreal winter (December–February) 1979–2006 as the initial basic state. We conclude, based on the fact that NAO onset is a nonlinear initial-value problem, that phase-strength asymmetry is an intrinsic characteristic of the NAO.

Optimal nonlinear excitation of decadal variability of the North Atlantic thermohaline circulation

Nonlinear development of salinity perturbations in the Atlantic thermohaline circulation (THC) is investigated with a three-dimensional ocean circulation model, using the conditional nonlinear optimal perturbation method. The results show two types of optimal initial perturbations of sea surface salinity, one associated with freshwater and the other with salinity. Both types of perturbations excite decadal variability of the THC. Under the same amplitude of initial perturbation, the decadal variation induced by the freshwater perturbation is much stronger than that by the salinity perturbation, suggesting that the THC is more sensitive to freshwater than salinity perturbation. As the amplitude of initial perturbation increases, the decadal variations become stronger for both perturbations. For salinity perturbations, recovery time of the THC to return to steady state gradually saturates with increasing amplitude, whereas this recovery time increases remarkably for freshwater perturbations. A nonlinear (advective) feedback between density and velocity anomalies is proposed to explain these characteristics of decadal variability excitation. The results are consistent with previous ones from simple box models, and highlight the importance of nonlinear feedback in decadal THC variability.

A Study of the North Atlantic Oscillation Using Conditional Nonlinear Optimal Perturbation

AbstractThe conditional nonlinear optimal perturbation (CNOP) method is used to explore the optimal precursors that trigger the North Atlantic Oscillation (NAO) anomaly pattern with a triangular T21, three-level, quasi-geostrophic global spectral model based on a viewpoint that the NAO is a nonlinear initial-value problem. With a three-dimensional winter climatological flow as the basic state, initially baroclinic localized optimal precursors on the northward flanks of the climatological Atlantic jet undergo wave breaking during their evolution into the NAO-like anomalies. Accompanied with the formation of the NAO, the north–south variability of the zonal mean westerly anomaly has arisen.Analysis reveals that in the evolution of optimal precursors, the role played by the self-interaction of perturbations (viz., the nonlinear process) in the onset of the negative-phase NAO (NAO−) event is stronger than that in the onset of the positive-phase NAO (NAO+) event. Both the perturbation/basic-state interaction and self-interaction of perturbations determine whether the NAO− event occurs, whereas the nonlinearity process in the NAO+ onset only appears to modulate the structure of the perturbation to have a dipole mode over the North Atlantic at the optimization time, and meanwhile cause this dipole mode to become zonally extended. That is to say, the nonlinear process indeed plays an important role during the onset of an NAO event and the CNOP method is a useful tool to identify the dynamics of the onset of NAO teleconnection patterns.

The Research on the Lake Eutrophication with CNOP Method

Using a dynamical model for nutrient cycling in shallow lakes, the approach of conditional nonlinear optimal perturbation (CNOP) was adopted to investigate the instability and the sensitivity of the lake ecosystem to finite-amplitude perturbations both related to the initial condition and the parameter. The results show that the ecosystem can be transformed from an oligotrophic (eutrophic) state to an eutrophic (oligotrophic) state with a CNOP as the perturbation, no matter how large the nutrient loading rate is. Above all, with the same restraints related to the initial perturbation and the parameter perturbation, CNOP has the largest effect on the lake ecosystem, which may be helpful to govern the lake ecosystem.

The time and regime dependencies of sensitive areas for tropical cyclone prediction using the CNOP method

This study examines the time and regime dependencies of sensitive areas identified by the conditional nonlinear optimal perturbation (CNOP) method for forecasts of two typhoons. Typhoon Meari (2004) was weakly nonlinear and is herein referred to as the linear case, while Typhoon Matsa (2005) was strongly nonlinear and is herein referred to as the nonlinear case.