Description

The XRO is an eXtended nonlinear Recharge Oscillator model for El Niño-Southern Oscillation (ENSO) and other modes of variability in the global oceans. It builds on the legacies of the Hasselmann stochastic climate model capturing upper ocean memory in SST variability, and the recharge oscillator model for the oscillatory core dynamics of ENSO. It constitutes a parsimonious representation of the climate system in a reduced variable and parameter space that still captures the essential dynamics of interconnected global climate variability.

This is updated version of XRO, with the model equations can be written as:

$$ \frac{d}{dt} \begin{pmatrix} X_{\text{ENSO}} \ X_M \end{pmatrix} = L \begin{pmatrix} X_{\text{ENSO}} \ X_M \end{pmatrix} + \begin{pmatrix} N_{\text{ENSO}} \ N_M \end{pmatrix} + \sigma_{\xi} G \xi, \quad (1) $$

$$ \frac{d\xi}{dt} = -r_{\xi} \xi + w(t), \quad (2) $$

where

$$ X_{\text{ENSO}} = [T_{\text{ENSO}}, h ] $$

and

$$ X_M = [T_{\text{NPMM}}, T_{\text{SPMM}}, T_{\text{IOB}}, T_{\text{IOD}}, T_{\text{SIOD}}, T_{\text{TNA}}, T_{\text{ATL3}}, T_{\text{SASD}}] $$

are state vectors of ENSO and other climate modes, respectively. This model allows for two-way interactions between ENSO and the other modes. $L$, $N$, $G(X)$ describe linear, nonlinear, multiplicative noise dynamics. For the detailed formulation of XRO model, please refer to our paper Zhao et al. (2024)[1] in Nature (v0.1).

This repository hosts the python package for XRO model. We have designed XRO to be user-friendly, aiming to be a valuable tool not only for research but also for operational forecasting and as an educational resource in the classroom. We hope that XRO proves to be both a practical and accessible tool that enhances your research and teaching experiences.

If you encounter problems in running XRO or have questions, please feel free to contact Sen Zhao (zhaos@hawaii.edu).

Installation

You can install XRO in two ways:

From PyPI

pip install XRO

From GitHub (latest development version)

pip install git+https://github.com/senclimate/XRO.git

Quick Start

XRO_Cookbook.ipynb is a Jupyter Notebook showing how to use XRO and reproduce the analysis of Zhao et al. 2024[1]. To successfully run the example, these open-source python modules may be necessary: numpy, xarray, climpred, matplotlib, and datetime.

XRO_realtime_states_forecasts.ipynb is a Jupyter Notebook showing how to calculate latest XRO state vectors from ORAS5 and conduct the realtime forecasts. Opertional XRO forecasts can be found at https://senzhao.netlify.app/climate/xro/.

XRO functionalities

XRO model is implemented in python with dependencies on only numpy and xarray. Key functionalities include:

• XRO.fit_matrix is a precedure to train nonlinear XRO parameters from the observational and climate model outputs
• XRO.simulate is a precedure to perform stochastic simulations with the trained parameters
• XRO.reforecastis a precedue to perform reforecasting or forecasting using the trained parameters and initial condictions

Additional helper utilities are provided for data preprocessing and statistical analysis, including:

• calc_XRO_indices – Compute the climate mode indices required by the XRO model (e.g., Niño3.4, WWV, NPMM, IOD, ATL3).
• detrend – Remove linear or quadratic trends from xarray.DataArray or xarray.Dataset objects.
• skew – Compute the skewness of climate variables.
• kurt – Compute the kurtosis of climate variables.
• pmtm - Thomson’s multitaper power spectral density (PSD) estimate
• xcorr - Cross correlations
• other functions can be found at stats.py state.py and visual.py

Applications

• The repository Recharge Oscillator (RO) Practical for the ENSO Winter School 2025. The practical covers theoretical and computational aspects of the RO framework, its applications in ENSO simulations, and forecasting. When other climate modes are not considered, the XRO simplifies to the RO, making it well-suited for use in this practical context.

Acknowledgement

Kindly requested to cite our paper Zhao et al. (2024) [1] and code [2] if use the XRO model in your published works.

References

[1] Zhao, S., Jin, F.-F., Stuecker, M.F., Thompson, P.R., Kug, J.-S., McPhaden, M.J., Cane, M.A., Wittenberg, A.T., Cai, W., (2024). Explainable El Niño predictability from climate mode interactions. Nature. https://doi.org/10.1038/s41586-024-07534-6

[2] Zhao, S. (2024). Extended nonlinear Recharge Oscillator (XRO) model for "Explainable El Niño predictability from climate mode interactions". Zenodo. https://doi.org/10.5281/zenodo.10681114