The Geometry of Axisymmetric Ideal Fluid Flows with Swirl

The sectional curvature of the volume preserving diffeomorphism group of a Riemannian manifold $M$ can give information about the stability of inviscid, incompressible fluid flows on $M$. We demonstrate that the submanifold of the volumorphism group of the solid flat torus generated by axisymmetric fluid flows with swirl, denoted by $\mathcal{D}_{\mu,E}(M)$, has positive sectional curvature in every section containing the field $X = u(r)\partial_\theta$ iff $\partial_r(ru^2)>0$. This is in sharp contrast to the situation on $\mathcal{D}_{\mu}(M)$, where only Killing fields $X$ have nonnegative sectional curvature in all sections containing it. We also show that this criterion guarantees the existence of conjugate points on $\mathcal{D}_{\mu,E}(M)$ along the geodesic defined by $X$.Comment: 8 page

The Production Rate of SN Ia Events in Globular Clusters

In globular clusters, dynamical evolution produces luminous X-ray emitting binaries at a rate about 200 times greater than in the field. If globular clusters also produce SNe Ia at a high rate, it would account for much of the SN Ia events in early type galaxies and provide insight into their formation. Here we use archival HST images of nearby galaxies that have hosted SNe Ia to examine the rate at which globular clusters produce these events. The location of the SN Ia is registered on an HST image obtained before the event or after the supernova faded. Of the 36 nearby galaxies examined, 21 had sufficiently good data to search for globular cluster hosts. None of the 21 supernovae have a definite globular cluster counterpart, although there are some ambiguous cases. This places an upper limit to the enhancement rate of SN Ia production in globular clusters of about 42 at the 95% confidence level, which is an order of magnitude lower than the enhancement rate for luminous X-ray binaries. Even if all of the ambiguous cases are considered as having a globular cluster counterpart, the upper bound for the enhancement rate is 82 at the 95% confidence level, excluding an enhancement rate of 200. Barring unforeseen selection effects, we conclude that globular clusters are not responsible for producing a significant fraction of the SN Ia events in early-type galaxies.Comment: 28 pages, 5 figures; ApJ submitte

The Diffeomorphism Group Approach to Vorticity Model Equations

In fluid mechanics, the vorticity provides a valuable alternative perspective of the behavior of flows. Constantin-Lax-Majda approached studying the 3D vorticity equation by proposing a 1D model equation with significant analytic similarities, the Constantin-Lax-Majda equation. This has been followed by a collection of model equations in both 1D and 2D whose behaviors capture many aspects of the full 3D equations. This thesis contains many new results for several of these equations. We begin by outlining the original analytic theory as well as the Euler-Arnold theory which studies these equations as geodesic equations on infinite dimensional manifolds. We build on the work of Castro-Cόrdoba and Bauer-Kolev-Preston to show that every solution to the Wunsch equation, a special case of the generalized Constantin-Lax-Majda equation, blows up in finite time. This result also applies to the Constantin-Lax-Majda equation itself. We also investigate the Euler-Weil-Petersson equation which has significant links to the Wunsch equation in the context of Teichmüller theory. Additionally, we lay the foundations for a geometric theory of the surface quasi-geostrophic equation (SQG). Originally discovered in the context of geophysical fluid mechanics (see Pedlosky), SQG was proposed by Constantin-Majda-Tabak as a 2D version of the 1D Constantin-Lax-Majda equation. In a blog post, Tao showed that SQG arises as the critical point of a functional. This discovery naturally leads to the formulation of SQG as an Euler-Arnold equation. In this thesis we show that the associated geometric space has a smooth, non-Fredholm Riemannian exponential map, and has arbitrarily large curvature of both signs. Finally we discuss the geometric setting for the Axi-symmetric Euler equations. Here we consider a 3D analogue of the 2D flows considered in Preston. Surprisingly, while the 2D flows exhibit negative curvature, we show that the corresponding 3D flows exhibit positive curvature and a rich structure of conjugate points. Such a result may have significant ramifications for our understanding of the nature of stability in 2D and 3D fluids