PhD, August 2004
Advisors: Keith Julien, Jeff Weiss
thesis.pdf
The Quasigeostrophic (QG) equations are an important model for rotationally constrained flow in our atmosphere and oceans. In the QG regime, turbulent flow is dominated by coherent vortices. These vortices are observed in the ocean, the atmospheres of giant planets and possibly in astrophysical disks. In two-dimensional flow the only important interaction is the merger of two same-sign vortices. In three-dimensional flow vortices may either merge or align vertically. The vertical alignment of coherent vortices is apparent in numerical simulations where it leads to the formation of large vertical scale features. In this thesis we study the processes of three-dimensional vortex merger and alignment using reduced models of vortex interaction and numerical simulations.
First, we derive a Hamiltonian model for the interaction of separate ellipsoidal vortices which we then use to study merger and alignment. A boundary for the region where vortex merger occurs is found and a similar boundary is shown to not exist for vortex alignment. Second, the numerical model is also used to study the interaction of ellipsoidal vortices in the full quasigeostrophic equations. The prediction of the reduced model is shown to be quite accurate for the region where vortex merger occurs. Vortex alignment is shown to be a much weaker effect than vortex merger and possible causes of alignment are suggested by the results of the simulations. Finally, a novel model for ocean turbulence is investigated in the situation of freely decaying turbulence. The inverse cascade of energy is studied and shown to slow down at larger scales, possibly explaining the buildup of energy at the scale of the Rossby deformation radius in the ocean.