Parametric rolling and green 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 green water, 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.

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 deck green water. 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 green water volume 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.

The study produced three key findings. First, parametric rolling significantly broadens the wave frequency range in which green 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 green 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, green 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.

This study establishes a reliable framework for analyzing the coupling effects between parametric rolling and green water. The results demonstrate that these two phenomena reinforce each other: parametric rolling broadens the occurrence range and green water volume, while deck green water 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.

Parametric rolling is a typical mode of dynamic instability, characterized by strong nonlinear dynamics and random wave excitation forces.International maritime organization (IMO) has issued the "Guidelines for assessment of the second generation intact stability of ships", which outlines the assessment methods and technical requirements. Predicting dynamic stability of ships under complex sea conditions is a key scientific challenge in the field of maritime safety. Stability criteria based on deterministic waves cannot accurately identify the instability domain in random wave environments.

In this paper, a single-degree-of-freedom equation for roll motion is established, incorporating both external excitation and parametric excitation, with nonlinear damping and restoring forces. Assuming that the ship's pitch and heave motions in waves are quasi-static processes, the roll restoring moment is calculated numerically using the strip theory. The righting arm is approximated with a polynomial expression for various wave directions, accurately capturing the characteristics of roll motion. The dynamic stability of nonlinear roll motion in random waves is investigated using a stochastic analysis method. The improved stochastic averaging method of energy envelope (ISAM-E) is introduced to account for the frequency component differences in rolling motion caused by both parametric and external excitation. ISAM-E is suitable for analyzing nonlinear roll motion under narrow-band spectra. Based on first-passage theory, the first-passage probability of stochastic roll motion is calculated under specified boundary and initial conditions.

Based on the first-passage probability approach, the C11 container ship is taken as an example to calculate the probabilities of the roll response exceeding 25° for full wave directions. The entire sea area is divided into three regions: high stability, medium stability, and low stability. This method effectively identifies the random sea conditions where the rolling response amplitude exceeds 25°.

The occurrence mechanism of parametric rolling and dynamic stability assessment are thoroughly explored through the application of the stochastic averaging method and first-passage theory. The methods proposed in this paper significantly enhance computational efficiency without considering non-ergodicity. This approach provides valuable insights into the stochastic stability of roll motion under various wave directions.