Glacier calving is a major contributor to sea-level rise, accounting for approximately 50% of mass loss from Greenland and Antarctic ice sheets. This process involves complex interactions between the ice, the glacier's internal stresses, and the ocean, including first-order processes like crevasse formation due to varying flow velocities, and second and third-order processes such as crack propagation and oceanic erosion. The calving process itself can have catastrophic consequences; falling or capsizing icebergs can generate significant tsunamis, posing serious threats to coastal communities, infrastructure, and ecosystems. In high-mountain proglacial lakes, these waves can even trigger devastating lake outburst floods. Existing numerical models for marine-terminating glaciers often focus on slow ice creep using continuum Eulerian methods, relying on simplified, empirically based calving laws. While some Lagrangian particle-based models exist that capture calving dynamics, they are computationally expensive and lack explicit water modeling, hindering the study of tsunami generation. Recent multiphysics finite element models incorporate oceanic melt effects but still fail to simulate the tsunamis themselves. Studies on landslide-induced tsunamis are abundant, but those specifically addressing calving-induced tsunamis are limited, and existing empirical equations often overestimate wave amplitudes. Previous work using foam-extend and the Immersed Boundary Method showed promise in reproducing wave characteristics but lacked a unified approach to simulate both fracture and tsunami generation. This research aims to address this gap by developing a model that explicitly simulates both ice fracture and hydraulic interactions.

Several numerical approaches have attempted to model glacier calving and associated tsunamis. Continuum Eulerian methods, like Elmer/Ice, primarily focus on slow ice creep, employing simplified empirical calving laws. Lagrangian particle-based models, while successfully reproducing some calving features and fractal debris size distributions, suffer from high computational cost and lack explicit water modeling, preventing tsunami simulation. Multiphysics finite element methods have shown progress by coupling slow glacier flow with a damage-based calving criterion, but they don’t simulate the resulting tsunamis. While research on landslide-induced tsunamis is extensive, the study of calving-induced tsunamis is relatively scarce, with empirical equations often proving inaccurate. A recent study using foam-extend and the Immersed Boundary Method successfully reproduced observed wave characteristics, but a comprehensive model capable of simulating fracture and tsunami formation simultaneously remained absent. This paper directly addresses this lack of a unified modelling approach.

The researchers developed a novel continuum damage Material Point Method (MPM) to model the coupled processes of glacier calving and tsunami generation. This hybrid Eulerian-Lagrangian approach uses Lagrangian material points to track mass, momentum, and deformation gradient, while a background Eulerian grid solves the mass and momentum conservation equations. The model incorporates a non-associative elastoplastic model for dynamic ice fracture based on the Cohesive Cam Clay (CCC) yield surface. This model employs a non-associative flow rule crucial for accurately representing the volume-preserving qualities of ice fracture, unlike associative flow rules used for more porous materials like snow. A softening law models dynamic ice fracture. Water is modeled using a nearly incompressible equation of state, preventing unrealistic volume changes. The mass and momentum balance equations are solved using the MPM, with the Affine Particle-In-Cell (APIC) method employed for transfer operations to ensure conservation of linear and angular momentum. The model was validated against both laboratory experiments and field observations. Laboratory experiments involved releasing ice-density blocks into a water tank to simulate different calving mechanisms (gravity-dominated fall, buoyancy-dominated fall, and capsizing). The model accurately reproduced wave amplitude and velocity in these experiments. A simple glacier calving geometry, for which analytical solutions exist, further validated the model's ability to simulate the interplay between buoyancy and ice weight, accurately predicting iceberg lengths. Finally, a real-world calving event at Eqip Sermia, Greenland, was simulated using the model, showing good agreement with field measurements of iceberg size, tsunami amplitude, and wave speed. The model's parameters were calibrated using field observations and data from previous studies on ice mechanics, carefully considering the uncertainty associated with the 2D to 3D wave amplitude transformation.

The developed MPM model successfully reproduced several key aspects of glacier calving and tsunami generation: 1. **Accurate Tsunami Simulation:** The model accurately replicated the amplitude and speed of tsunamis generated by various calving mechanisms (gravity-dominated fall, buoyancy-dominated fall, and capsizing) observed in large-scale laboratory experiments. Minor deviations in maximum wave amplitude were attributed to the empirical nature of the 2D/3D transformation process. 2. **Iceberg Size Prediction:** Simulations of a simplified glacier calving geometry, where analytical solutions exist, demonstrated the model's ability to predict iceberg size accurately, capturing the interplay between buoyancy and ice weight. The predicted iceberg lengths aligned well with predictions from a simple analytical bending model. 3. **Real-World Validation:** The model successfully simulated a past calving event at the Eqip Sermia glacier in Greenland. The simulated iceberg geometry, dimensions, tsunami amplitude, and wave speed showed good agreement with field observations, validating the model's applicability to real-world scenarios. The model correctly predicted a tsunami wave amplitude of 50 m close to the calving front and 3.3 m at a distant tide gauge, aligning well with measurements. 4. **Computational Efficiency:** The hybrid Eulerian-Lagrangian framework of the MPM significantly reduced computational time compared to discrete element methods, enabling the simulation of complex 3D calving events involving millions of particles within a reasonable timeframe on standard computing hardware. 3D simulations with 20-36 million particles were completed in less than a day on a standard workstation.

The findings of this study demonstrate the effectiveness of the developed MPM model in simulating the complex multiphase interaction between ice and water during glacier calving and tsunami generation. The successful replication of laboratory experiments and real-world observations validates the model’s accuracy and provides confidence in its ability to predict tsunami characteristics. The model's ability to accurately simulate iceberg size and tsunami parameters is a significant advancement over existing models that either rely on simplified calving laws or lack explicit water modeling. The improved accuracy and computational efficiency make this model ideal for hazard assessment and mitigation in coastal regions vulnerable to glacier-related tsunamis. The ability to simulate multiple calving events also enhances model validation and allows for a more comprehensive understanding of the dynamics involved. While the nearly incompressible water model works well, using computational fluid dynamics (CFD) to solve incompressible Navier-Stokes equations would yield potentially more accurate water dynamics. However, this would significantly increase computational cost and complexity. Future work could incorporate this improvement. Similarly, while the model captures many essential aspects of glacier calving and tsunami formation, considerations of factors like ice anisotropy and viscous creep could enhance the model’s predictive capabilities further.

This paper presents a novel continuum damage Material Point Method (MPM) model for simulating glacier calving and tsunami generation, successfully validated against both laboratory and field data. The model accurately predicts iceberg size, tsunami amplitude, and wave speed, offering significant improvement over existing methods. Its computational efficiency facilitates large-scale 3D simulations, making it a valuable tool for hazard assessment and mitigation strategies in coastal regions susceptible to glacier-related tsunamis. Future research can extend this model by incorporating more detailed ice rheology, accounting for ice anisotropy, and further refining the water model using CFD techniques.

The model's accuracy is influenced by the empirical nature of the 2D to 3D wave amplitude transformation. The model's current formulation assumes a relatively stiff material which is appropriate for ice but may not be suitable for all materials. While the model accurately simulates short-term brittle ice fracture, it does not account for viscous creep or complex basal processes affecting crevasse formation; future work could explore coupling this model with existing continuum glacier flow models to incorporate these long-term processes. Finally, the model's calibration relies on a selection of back-calculated parameters, highlighting potential uncertainty.