# Mathematical Studies of Wave Propagation in Sea-Ice

Abstract (Summary)

This thesis presents mathematical studies of wave propagation at the surface of the ocean covered with sea-ice sheets. During the winter months the sea around the coast of Antarctica freezes to form a vast area of ice covered region. The ice sheets found in this area are often large and apparently featureless. From these properties, a large scale sea-ice sheet has been modelled as a homogeneous thin elastic plate with a constant Young's modulus when the deformation of the ice sheet is small.
A partial differential equation describing the motion of an ice sheet is de-rived from the classical theory of linear elasticity. A well known system of partial differential equations is derived to describe the vertical deflection of a thin elastic plate coupled with a linearized Bernoulli's equation at the surface and Laplace's equation for incompressible fluid of finite depth. The Fourier transform is used to derive the vertical deflection of the ice sheet when a localized time harmonic vertical load is applied. A fundamental solution can be expressed by infinite summations of fractional functions at complex roots of the dispersion equation. The inverse Fourier transform is calculated by hand using the Hankel transform and the resulting formulae are directly turned into numerically stable computer codes without any modification. It is shown from the finite water depth solution, that the infinite series expansion of the deflection can be reduced to a sum of special functions whose modes are three roots of a fifth order polynomial. Furthermore, the space and time variables are non-dimensionalized using a characteristic length and characteristic time which are determined by the thickness and Young's modulus of the ice sheet. As a consequence of the non-dimensionalization, the solutions are insensitive to the ice thickness and categorized according to distinctive physical characteristics. It is conversely shown that the characteristic length and characteristic time can be measured from direct observations of the flexural motion, such as the surface strain or tilt, of the ice sheet.
Interaction between two semi-infinite ice sheets that are joined by a straight infinite line transition is considered. When a plane wave is obliquely incident on the transition, reflection and transmission of wave energy occurs. Analytical formulae for the coefficients of a modal expansion of the waves in the ice sheet are derived using the Wiener-Hopf technique and formulae for the reflection and transmission coefficients of the surface waves are also presented. The formulae are directly turned into stable computer codes. The application of the Wiener-Hopf technique given here is modified from a standard method to accommodate the incident wave from infinity better. A set of physical conditions at the transition can be expressed using a pair of 4 x 4 matrix and 4-element vector, whose elements are computed from the coefficients of the modal expansion of the solutions. Hence the solution procedure is able to cope with various transition conditions simply by changing the matrix and the vector.
Modelling of finite ice sheets using the boundary integral equation method is also considered. Formulation of boundary integral equations is given and it is shown that the dynamics of the ice sheet is represented by the boundary integrals only on the edge of the ice sheet. Connection between the Wiener-Hopf technique and boundary integral method is made.
Bibliographical Information:

Advisor:Dr. C. Fox

School:The University of Auckland / Te Whare Wananga o Tamaki Makaurau

School Location:New Zealand

Source Type:Master's Thesis

Keywords:

ISBN:

Date of Publication:01/01/2002