Diagnosing and predicting ENSO using cyclostationary linear inverse models with persistent stochastic forcing

Linear inverse models (LIMs) are widely used in climate science to diagnose dynamical relationships among climate variables and to predict large-scale variability such as El Niño-Southern Oscillation (ENSO). Recent extensions from stationary to cyclostationary LIMs (CS-LIMs) incorporate seasonally varying dynamics, but most formulations still assume white-noise forcing, which implicitly requires that the system state and the random forcing are well separated in frequency. This assumption limits their ability to represent realistic atmosphere-ocean interactions.In this study, we further advance the cyclostationary LIM framework by introducing Ornstein-Uhlenbeck colored noise to represent persistent atmospheric stochastic forcing. We refer to this extension as CS-Colored-LIM. Incorporating persistent noise enables a more physically consistent representation of unresolved atmospheric variability and its cumulative influence on the coupled system.We compare the newly developed CS-Colored-LIM and conventional LIMs in terms of their Niño 3.4 forecast skill and their ability to capture essential ENSO features and the influence of stochastic forcing. Our analysis demonstrates that CS-Colored-LIM accurately reproduces the seasonal cycle of ENSO variability, providing a framework for studying ENSO phase locking and the spring prediction barrier. Moreover, despite the submonthly characteristic timescale of persistent noise, its cumulative contribution to Niño 3.4 evolution exceeds 10%, revealing the non-trivial role of persistent stochastic forcing.Forecast experiments show that cyclostationary formulation improves short-range prediction skill (≤ 12 months) through better representation of month-to-month variability, while colored noise enhances longer-lead performance (>12 months) by accounting for persistent atmospheric forcing. CS-Colored-LIM benefits from both effects, yielding statistically significant improvements in correlation skill, more reliable ensemble forecasts, and enhanced prediction of major ENSO events, compared to conventional LIMs. Consequently, CS-Colored-LIM provides a simple yet powerful framework for long-range ENSO diagnosis and prediction, offering new insights into the interaction between seasonally varying dynamics and persistent stochastic forcing.