Spectral analysis of ice shelf vibrations

Abstract

Ice shelves are large floating ice plates that are attached to ice sheets or glaciers. They play a significant role in buttressing glaciers and supporting the stability of the glacier-ice shelf system. Ice shelves have the potential to contribute to rising sea levels in the coming years. It is important to understand the interactions between the ice shelf and the ocean. Therefore, studying the vibrations of the ice shelf may help us assess the potential risk and develop effective mitigation strategies. This study explores several connected topics related to the vibrations of ice shelves. It focuses on how different boundary conditions, ice shelf structures, and fluid environments affect their vibrational behaviour. More precisely, we want to study an ice shelf floating on shallow water (cavity) decoupled from the ocean. In this work, we aim to develop a method to calculate the vibration modes of the fluid (velocity potential) in the cavity and the ice shelf (vertical displacement). Using the mode expansion method, these mode shapes are expanded to gain insight into the behaviour of the ice shelf–cavity system. All solutions are obtained through a numerical method that simplifies all calculations to matrix multiplications. In this work, the ice shelf is modelled using linear plate theory, which leads to a fourth-order equation requiring four boundary conditions depending on the geometry of the ice shelf. We then apply the shallow water theory to derive the governing equation for the fluid, which is a sixth-order equation that requires six boundary conditions. Finding the angular frequencies is essential in our work, as each angular frequency is associated with a mode shape (eigenfunction). We use a numerical technique to find the singularities of the matrix. This technique is challenging, and the only approach is an exhaustive search. It should be noted that finding the modes is the most significant numerical challenge. Once the mode shapes of the fluid are determined, and using the benefit of the shallow water theory, the corresponding mode shapes of the ice shelf can be constructed accordingly. Moreover, this approach enables us to simulate vibrations in the time domain using specific initial conditions. The vibration solution of the system is first presented in one dimension using Cartesian coordinates, where the solution is expressed in terms of the exponential functions. The shape of the ice shelf is then extended to a circular, two-dimensional geometry in polar coordinates, where the solution is expressed in terms of Bessel and modified Bessel functions. Additionally, we consider the case of a circular plate with mixed boundary conditions as a stepping stone to understanding the case of an ice shelf, which naturally has different boundary conditions. This work provides valuable information on the ice shelf under different geometrical configurations and the fluid under various boundary conditions. These findings may help us improve our understanding of the ice shelf and its behaviour under certain conditions.