On the Bending Kinetic Energy and a Revisit to Cross-Scale Kinetic Energy Diagnostics

A comprehensive and accurate representation of kinetic energy ( KE ) is crucial for understanding multiscale processes in the Earth system. Traditional 2-reservoir frameworks, which partition total kinetic energy into mean kinetic energy ( MKE ) and eddy kinetic energy ( EKE ) components, inevitably result in an imbalanced energy cascade owing to the quadratic nature of KE and the non-negligible cross term in realistic flows. In this study, we introduce a new KE reservoir, termed “bending kinetic energy” ( BKE ), inspired by an analogy with the vibrational coupling of stretching and bending modes in the hydrogen bonds of water molecules. This leads to a physically consistent and objective diagnostic framework in which energy exchanges among MKE , EKE , and BKE are precisely balanced. In this framework, any loss in one reservoir is exactly offset by the corresponding gains in the other reservoirs, thereby eliminating the asymmetry observed in conventional 2-reservoir frameworks. We demonstrate the utility of this diagnostic framework through a case study of the Kuroshio Extension region, where inclusion of BKE reconciles observed discrepancies in the direction of the energy cascade. Overall, the proposed diagnostic framework offers a complete and unified description of the KE cycle, highlighting the critical role of BKE in closing the KE budget. The framework provides a valuable tool for advancing our understanding of multiscale energy cascades in the Earth system.