The El Niño–Southern Oscillation (ENSO) is a significant climate phenomenon that appears periodically in the tropical Pacific. The intermediate coupled ocean–atmosphere Zebiak–Cane (ZC) model is the first and classical one designed to numerically forecast the ENSO events. Traditionally, the conditional nonlinear optimal perturbation (CNOP) approach has been used to capture optimal precursors in practice. In this paper, based on state-of-the-art statistical machine learning techniques

Generally, the statistical machine learning techniques refer to the marriage of traditional optimization methods and statistical methods, or, say, stochastic optimization methods, where the iterative behavior is governed by the distribution instead of the point due to the attention of noise. Here, the sampling algorithm used in this paper is to numerically implement the stochastic gradient descent method, which takes the sample average to obtain the inaccurate gradient.

, we investigate the sampling algorithm proposed inIn the global climate system, the most prominent phenomenon of year-to-year fluctuations is the El Niño–Southern Oscillation (ENSO), which makes a huge impact on Earth’s ecosystems and human societies via influencing temperature and precipitation

Modern studies of the ENSO theory date back to the late sixties of the last century.

In numerical prediction, a key issue that we often meet is the short-term behavior of a predictive model with imperfect initial data. In other words, it is of vital importance to understand the sensitivity of the numerical models to errors in the initial data. The simplest and most practical way is to estimate the likely uncertainty for the initial data polluted by the most dangerous errors. Currently, the conventional approach to capture the optimal initial perturbation is the so-called conditional nonlinear optimal perturbation (CNOP) approach innovatively introduced in

CNOPs are often obtained by implementing nonlinear optimization methods, mainly including the spectral projected gradient (SPG) method

It is worth noting that the first-order optimization method employed to obtain the maximum in the scientific community of fluid mechanics is the method of Lagrange multipliers

The paper is organized as follows. Section

In this section, we first briefly describe the basic process to compute the optimal precursors by the use of the CNOP approach in the ZC model.

Although the CNOP approach has been extended to investigate the influences of boundary errors and model errors on atmospheric and oceanic models

Let

Throughout the paper, all vectors are denoted by bold italics.

where the indexNext, we consider the objective function that is on the initial errors. As our primary concern is maximizing the target quantity solely dependent on the nonlinear evolution state of the SST anomalies, we define the objective function as

Based on Stokes' formula,

The average of the function values (Eq.

After the CNOP approach is imported to the ZC model

The spatial patterns of the optimal precursors in terms of SST anomalies

Recall the optimal precursors bringing about the El Niño event, which is obtained by the CNOP approach in

The optimal precursors

We have shown considerable similarities in the spatial patterns of the optimal precursors bringing about the El Niño event in Fig.

Both the spatial patterns and the objective values indicate that the optimal precursor obtained through the sampling algorithm, using only 200 samples, is very similar to the one obtained through the baseline adjoint algorithm. To show the efficiency of the sampling method, a comparison of computation times is necessary. As mentioned in Sect.

The comparison of computation times between the adjoint method and the sampling method under the parallel computation. The Fortran code was run on the following CPU: Intel^{®} Xeon^{®} Gold 6132 Processor, 19.25M Cache, 2.60 GHz, with eight nodes and 28 cores per node.

It needs to take about

Based on the CNOP approach, the statically spatial patterns of the optimal precursors of ENSO events are in terms of both SST anomalies and thermocline depth anomalies. In Sect.

The spatial patterns of the nonlinear time evolution of the optimal precursors in terms of SST anomalies.

Recall the nonlinear time evolution of SST anomalies simulated by the coupled ocean–atmosphere ZC model shown in

Based on the nonlinear time evolution of SST anomalies simulated in Fig.

Currently, the main variable that is considered from the ENSO forecasts of the coupled climate models is the Niño 3.4 SST anomaly index, which is used by the National Climate Centre (NCC) in Australia to classify ENSO conditions. Here, we show that the Niño 3.4 SST anomaly indices change nonlinearly along the time evolution line within a model year in Fig.

Here, we can find that the relative Niño 3.4 SST anomaly index from the sampling method with

Based on state-of-the-art statistical machine learning techniques, the sampling method to compute CNOPs is proposed in

The nonlinear time evolution of the Niño 3.4 SST anomaly index within a model year. The bars represent the range of errors obtained from running the sampling method

The nonlinear time evolution of relative Niño 3.4 SST anomaly index within a model year (the Niño 3.4 SST anomaly index obtained by the sampling method minus that by the baseline adjoint method). The bars represent the range of errors obtained from running the sampling method

For a realistic global climate system model (GSCM) or atmosphere–ocean general circulation model (AOGCM), it is often impractical to develop the adjoint model, so the sampling method provides a probable way of computing CNOPs to investigate its predictability. An interesting direction for further research is to investigate CNOPs computed by the sampling method in the numerical models that are used in realistic prediction and forecast, such as the Weather Research and Forecasting (WRF) model, a state-of-the-art mesoscale numerical weather prediction system for operational forecasting applications. Another interesting direction is to attempt to use the sampling method to realize a more (or less) nonlinearly stable flow by changing some aspect of the system

We use the codes of

BS constructed the basic idea of this paper, derived all formulas, and wrote the paper. JM coded the sampling method in the ZC model in Fortran (figures drawn by Python) and joined the discussions of this paper. Both authors contributed to the writing of the paper.

The contact author has declared that neither of the authors has any competing interests.

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Junjie Ma participated in this work during the final semester of his PhD under the supervision of Wansuo Duan at the Institute of Atmospheric Physics, Chinese Academy of Sciences.

This research has been supported by the National Natural Science Foundation of China (grant no. 12241105) and the Bureau of Frontier Sciences and Education, Chinese Academy of Sciences (grant no. YSBR-034).

This paper was edited by Pierre Tandeo and reviewed by two anonymous referees.