Objectives Parametric rolling and water on deck are two critical nonlinear phenomena closely associated with ship motion stability, especially in severe sea conditions. Their concurrent occurrence, often triggered by overlapping environmental conditions, can significantly exacerbate ship instability and even lead to capsizing. Existing studies have primarily focused on individual phenomena or relied on model tests and computational fluid dynamics (CFD) methods, which lack analysis about their coupling mechanism. Thus, this research aims to investigate the interaction between parametric rolling and water on deck, quantify their mutual influences, and reveal the coupling effects on ship nonlinear motion and stability. The results are expected to fill the gap in mathematical modeling of the two coupled phenomena.

Methods A three-degree-of-freedom (3-DOF) coupling model integrating heave, pitch, and parametric rolling was established within the framework of potential flow theory, explicitly incorporating the effects of green water on deck. The model accounts for three key factors: additional inclining moments induced by water accumulation, changes in ship displacement and center of gravity, and changes of the wetted surface due to altered floating conditions. To solve the model, the 1.5-degree-of-freedom concept was adopted to calculate the roll restoring force, considering the unidirectional coupling of heave and pitch on rolling. The volume of water on deck was computed by integrating inflow and outflow velocities along the deck edge perimeter, with quasi-static equilibrium assumed for water distribution. CFD simulations using overlapping grids and the Volume of Fluid (VOF) method were conducted for validation, with the C11 container ship selected as the study object. Then the model results were cross-validated against CFD data under different wave steepness conditions.

Results The research yielded three key findings. Firstly, parametric rolling significantly expands the wave frequency range prone to water on deck. For instance, at the wave height of 7.86 m, water on deck only occurred when parametric rolling was considered (wave frequency: 0.433–0.485 rad/s), while no overtopping was observed without accounting for this motion. At a higher wave height of 13.1 m, the frequency range for water on deck expanded from 0.327–0.433 rad/s (without parametric rolling) to 0.327–0.512 rad/s (with parametric rolling), accompanied by a substantial increase in overtopping volume. Secondly, water on deck and accumulated water exert a strong nonlinear impact on the righting arm (GZ) curve. The GZ value decreases with increasing water volume, even becoming negative at small roll angles, which significantly impairs wave stability. This effect is more pronounced when wave troughs are at the midship. And the negative region of GZ expands as wave steepness increases. Thirdly, when there are bulwarks on the deck, water periodically flows in and out, maintaining a constant water accumulation. This leads to increased ship draft and a slight bow down angle, amplifying parametric rolling response. The amplitude of parametric rolling increased by 5.74% as the bulwark height rose from 0 m to 3 m, with the effect intensifying as accumulated water volume increased. Cross-validation with CFD results showed that the model’s predictions for overtopping volume and rolling amplitude had errors within 11% and 10%, respectively, confirming its reliability.

Conclusions This study establishes a reliable framework for investigating the coupling effects of parametric rolling and water on deck. The results demonstrate that the two phenomena interact synergistically: parametric rolling broadens the occurrence range and greenwater volume of water on deck, while water on deck reduces ship stability and enhances parametric rolling response. The coupling mechanism, characterized by altered GZ curves, displacement, and floating conditions, highlights the necessity of considering their mutual effects in ship stability assessments. The proposed model and findings provide valuable references for improving safety measures to mitigate risks associated with nonlinear motions in severe sea conditions. Future research will focus on incorporating dynamic water flow effects and roll damping changes to further refine the model’s accuracy.