My thesis work consisted of several parts: A slanted quasi-geostrophic (QG) model used to study geophysical turbulence; a study of vortex cores, circulation cells, and filaments in QG turbulence; and a protoplanetary disk model of the gas that surrounds young stars.
Slanted QG
The quasi-geostrophic (QG) potential vorticity equation is a reduced model for stratified, rotating flows which is often employed in atmospheric and ocean dynamics. QG, first derived by Charney in 1948, has been successful in analytic models of Rossby waves, cyclogenesis, and ocean basin circulation, as well as fundamental numerical models.
A weakness of QG is that the Rossby number must be small (i.e. rotation is stronger than advection), so it is only applicable at mid- and high-latitudes. I worked on an alternate asymptotic derivation of the primitive equations dubbed "Slanted QG" where the full rotation vector is kept, not just the vertical component of rotation as in standard QG. This may be an appropriate model to study the dynamics of stratified flows near the equator. I created a 3D periodic pseudo-spectral numerical model of slanted QG to study vortex behavior and the energy cascade in this regime.
Isotropic decay in quasigeostrophic turbulence (13 MB, .avi)
Protoplanetary Disks
Protoplanetary disk models are used to study Rossby waves, shear instabilities, and vortex formation in gas disks around young stars that eventually lead to planetary formation. In collaboration with Glen Stewart (LASP) we are studying a reduced, coupled system for vorticity and temperature which was derived from the anelastic equations. Linear analysis of this system shows that the conditions of radiative damping and a radial temperature gradient results in baroclinic instabilities which lead to vortices in the disk. These vortices efficiently concentrate gas and dust and are a potential mechanism for planetary formation. I have created a numerical model of this reduced equation set which is pseudo-spectral (Fourier-Chebyshev basis functions) on a 2D annulus. We use these analytical and numerical tools to study the physical mechanisms that drive vortex formation in protoplanetary disks.
Protoplanetary disk simulation showing how
anticyclonic vortices are stable and long-lived, while cyclonic ones are
not. (13 MB, .avi)