Objectives Parametric rolling and water on deck are two critical nonlinear phenomena closely related to ship motion stability, particularly under severe sea conditions. Their concurrent occurrence, often triggered by overlapping environmental factors, can greatly amplify ship instability and even lead to capsizing. Existing studies have predominantly examined these phenomena in isolation or relied on model experiments and computational fluid dynamics (CFD) simulations, which leaves a gap in understanding their coupled mechanisms. Therefore, this research aims to investigate the interaction between parametric rolling and water on deck, quantify their mutual influence, and reveal their combined effects on ship nonlinear motion and stability. The findings are expected to fill the gap in mathematical modeling of the two coupled phenomena.

Methods A three-degree-of-freedom (3-DOF) coupling model incorporating heave, pitch, and parametric roll motions was developed within the framework of potential flow theory, explicitly accounting for the effects of water on deck. The model captures three key factors: additional inclining moments induced by water accumulation, variations in ship displacement and center-of-gravity position, and changes in the wetted surface resulting from altered floating conditions. To solve the model, the 1.5-degree-of-freedom model was employed to calculate the roll restoring force, accounting for the unidirectional coupling effects of heave and pitch on roll motion. The volume of water on deck was determined by integrating inflow and outflow velocities along the deck edge perimeter, assuming a quasi-static distribution of the accumulated water. CFD simulations were performed using overlapping grids combined with the volume of fluid (VOF) method, with the C11 container ship selected as the study object. The model results were then cross-validated against CFD data under different wave steepness conditions.

Results The study produced three key findings. First, parametric rolling significantly broadens the wave frequency range in which water on deck occurs. For instance, at the wave height of 7.86 m, overtopping occurred only when parametric rolling was included in the model (wave frequency range of 0.433–0.485 rad/s), whereas no water on deck was observed when parametric rolling was excluded. At a higher wave height of 13.1 m, the overtopping frequency range increased from 0.327–0.433 rad/s (without parametric rolling) to 0.327–0.512 rad/s (with parametric rolling), accompanied by a substantial rise in overtopping volume. Second, water on deck and the accumulated water load induce strong nonlinear effects on the righting arm (GZ) curve. The GZ value decreases with increasing water volume, even becoming negative at small roll angles, which significantly impairs a ship's stability in waves. This effect is more pronounced when the wave trough is located at the midship, and the range of negative GZ expands with increasing wave steepness. Third, when bulwarks are present, water periodically flows in and out of the deck, resulting in persistent water accumulation. This leads to an increased draft and a slight bow-down trim, which in turn amplifies the parametric rolling response. The roll amplitude increased by 5.74% as the bulwark height increased from 0 m to 3 m, with the amplification effect becoming stronger as the accumulated water volume increased. Cross-validation with CFD simulations demonstrated errors within 11% for overtopping volume and 10% for roll amplitude, confirming the model's reliability.

Conclusions This study establishes a reliable framework for analyzing the coupling effects between parametric rolling and water on deck. The results demonstrate that these two phenomena reinforce each other: parametric rolling broadens the occurrence range and volume of water on deck, while water on deck reduces ship stability and intensifies the parametric rolling response. The coupling mechanism, manifested through altered GZ curves, displacement, and floatation conditions, highlights the importance of accounting for their mutual effects in ship stability assessments. The proposed model and findings provide valuable references for improving safety measures aimed at mitigating risks associated with severe nonlinear ship motions in harsh sea states. Future research will focus on incorporating dynamic water flow effects and roll damping changes to further improve the model fidelity.