Abstract:

Geophysical fluid dynamics is the study of fluids on the sphere in which the Coriolis force plays an important role, and is of great interest and importance, both theoretically and practically, as the foundation of modern weather and climate modeling. Many problems in geophysical fluid dynamics can be formulated in a way to take advantage of convolutions and fast summation techniques, which are methods for approximating integral transforms quickly.

The thesis starts by presenting a Cubed Sphere Fast Multipole Method (CSFMM) that is suitable for $O(N)$ fast summation on the sphere, adapting techniques from the Barycentric Lagrange Tree Code and the Barycentric Lagrange Dual Tree Traversal Fast Multipole Method for use for problems with spherical geometry, and showing a number of speed up and error results, demonstrating that the CSFMM is both fast and accurate for a variety of different problems. This technique is then applied to three different problems.

The first problem is that of computing Self Attraction and Loading in ocean models, an important term encompassing physical effects relating to the gravitation of water and the elastic deformation of the Earth. The computation of Self Attraction and Loading has been challenging in the past, mainly being computed using a scalar approximation or using spherical harmonics. This thesis demonstrates a new technique for computing the Self Attraction and Loading by deriving a new integral kernel for the problem, before then discussing the implementation of this convolution and CSFMM in the Modular Ocean Model.

The next problem is that of solving the Barotropic Vorticity Equation with a Lagrangian Particle Method. This equation describes the conservation of potential vorticity for a two dimensional incompressible inviscid fluid on a rotating sphere. The fluid velocity can be related to the vorticity through a Biot-Savart law, and when discretized using a Lagrangian particle Method, the dynamics naturally admit a formulation as a $N$-body problem, to which we apply the CSFMM. The accuracy and speed of this technique is tested, before using the method to explore a variety of problems.

Lastly, the previous solver is extended to work for the Shallow Water Equations, an equation set which in addition to vorticity effects, also allows for fluid divergence. For this problem, the Biot-Savart law is more complicated, incorporating both vorticity and divergence. This solver is then tested on a range of test cases to check for accuracy. Additionally, this solver is designed for portability, including with graphical processing units, allowing for significant speedups.