Modelling vortex-vortex and vortex-boundary interaction

The motion of two-dimensional inviscid, incompressible fluid with regions of constant vorticity is studied for three classes of geophysically motivated problem. First, equilibria consisting of point vortices located near a vorticity interface generated by a shear flow are found analytically in the linear (small-amplitude) limit and then numerically for the fully nonlinear problem. The equilibria considered are mainly periodic in nature and it is found that an array of equilibrium shapes exist. Numerical equilibria agree well with those predicted by linear theory when the amplitude of the waves at the interface is small. The next problem considered is the time-dependent interaction of a point vortex with a single vorticity jump separating regions of opposite signed vorticity on the surface of a sphere. Initially, small amplitude interfacial waves are generated where linear theory is applicable. It is found that a point vortex in a region of same signed vorticity initially moves away from the interface and a point vortex in a region of opposite signed vortex moves towards it. Configurations with weak vortices sufficiently far from the interface then undergo meridional oscillation whilst precessing about the sphere. A vortex at a pole in a region of same sign vorticity is a stable equilibrium whereas in a region of opposite-signed vorticity it is an unstable equilibrium. Numerical computations using contour dynamics confirm these results and nonlinear cases are examined. Finally, techniques based on conformal mapping and the numerical method of contour dynamics are presented for computing the motion of a finite area patch of constant vorticity on a sphere and on the surface of a cylinder in the presence of impenetrable boundaries. Several examples of impenetrable boundaries are considered including a spherical cap, longitudinal wedge, half-longitudinal wedge, and a thin barrier with one and two gaps in the case of the sphere, and a thin island and ‘picket’ fence in the case of the cylinder. Finite area patch motion is compared to exact point vortex trajectories and good agreement is found between the point vortex trajectories and the centroid motion of finite area patches when the patch remains close to circular. More exotic motion of the finite area patches on the sphere, particularly in the thin barrier case, is then examined. In the case when background flow owing to a dipole located on the barrier is present, the vortex path is pushed close to one of the barrier edges, leading to vortex shedding and possible splitting and, in certain cases, to a quasi-steady trapped vortex. A family of vortex equilibria bounded between the gap in the thin barrier on a sphere is also computed. Acknowledgements I would first and foremost like to express my heartfelt gratitude to my supervisor, Robb McDonald, for all the help and guidance he has given me throughout my time at UCL. It has been a pleasure to work with him and learn from his expertise, and I thank him for playing a big part in making my time at UCL so enjoyable. I would like to thank the Geophysical Fluid Dynamics group for the insightful conversations and their input and in particular my predecessor, Alan Hinds, for all his help and suggestions when I began my PhD. I am indebted to the UCL graduate school for their financial assistance that has allowed me to attend some prestigious conferences during my studies and of course, I owe a great deal of gratitude to my funding body, the Natural Environment Research Council. I would also like to thank my fellow PhD students who have created such a pleasant environment at UCL, and in particular the students with whom I began my PhD, Tommy, Jo, Brian and Isidoros. I owe a big thank you to all of the administrative staff who have been so helpful throughout my PhD. I also wish to warmly acknowledge the support of all of my friends and I would like to thank Guy and Phil for the great holidays during my PhD and Will and Francois for all of the entertainment in and around London. I have a giant thank you to say to my girlfriend, Chierh, for her patience, kindness and warmth (and for all the great meals that have kept me fit and healthy over the past couple of weeks). Yn olaf, hoffwn diolch fy teulu; fy Mam, Dad, Brawd, Chwaer ac fy Nain diffoddiedig, am eu cariad a cymorth trwy gydol fy addysg. Hoffwn diolch fy Mam am eu geiriau o cymorth yn y misoedd diweddar.