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Realizing Surface Driven Flows in the Primitive Equations

Abstract

The surface quasi-geostrophic (SQG) model describes the evolution ofÌýbuoyancy at vertical boundaries in the limit of infinitesimal RossbyÌýnumber. In this regime, the quasi-geostrophic approximations areÌýexpected to hold. Numerical simulation of the SQG model often generateÌýsmall-scale vortices which may have Rossby numbers that approach unityÌýand may be outside the range of SQG. In this thesis we investigate theÌýevolution of a surface trapped elliptical vortex in both the SQG modelÌýand the non-hydrostatic Boussinesq primitive equations (PE) which areÌýbetter able to describe a wider range of oceanic dynamics. Thus, inÌýthe PE, we can vary the Rossby number in order to understand how theÌýsurface trapped vortex breaks down at the smaller-scale during its evolution. For small Rossby number, we confirm that the PE match the SQG prediction very well. For larger Rossby number however, we find that the models do not agree and different dynamics begin emerging in the PE. In particular, we find that the thin filament instability in the surface buoyancy field, common to SQG, begins to stabilize as theÌýRossby number increases and thus the emergence of the secondaryÌýsmall-scale vortices is halted. The core of the vortex spreads out andÌýbecomes much more uniform for larger Rossby number. The energyÌýspectrum of the surface trapped vortex steepens from a power law ofÌý-5/3 to about -3 and the divergent energy grows as the Rossby numberÌýapproaches unity. The growing divergent energy is an indication thatÌýinertia-gravity waves are generated in the simulation and we do indeedÌýobserve these in the vertical velocity field. We conclude that whenÌýthe Rossby number of the surface trapped elliptical vortex is at leastÌý0.05 new dynamics emerge and the PE must be used to attain an accurateÌýdescription of the evolution of the flow.