Climate‐Adaptive and Cascade‐Constrained Machine Learning Prediction for Sea Surface Height Under Greenhouse Warming

Machine learning (ML) has achieved remarkable success in climate and marine science. Given that greenhouse warming fundamentally reshapes ocean conditions such as stratification, circulation patterns and eddy activity, evaluating the climate adaptability of the ML models is crucial. While physical constraints have been shown to enhance the performance of ML models, kinetic energy (KE) cascade has not been used as a constraint despite its importance in regulating multi-scale ocean motions. Here we present two sea surface height (SSH) prediction models (with and without KE cascade constraint) and quantify their climate adaptability in the Kuroshio Extension. Both models exhibit only slight performance degradation under greenhouse warming conditions. Incorporating the KE cascade as a physical constraint significantly improves the model performance, reducing eddy kinetic energy errors by 19.5% in the present climate and 19.6% under greenhouse warming conditions. Additional validations using satellite observations and in the Gulf Stream region further confirm the robustness of the proposed models. Compared with the traditional KE spectrum constraint, the model with the KE cascade constraint maintains improvements in the spectrum of KE while yielding more than 14% improvements in the cross-scale transfer of KE. This work presents the first application of the KE cascade as a physical constraint for ML-based ocean state prediction and demonstrates its robust adaptability across climates, offering guidance for the further development of global ML models for both present and future conditions.