The Fundamental Gap For Hyperbolic Triangles

In 1983, M. van den Berg made his Fundamental Gap Conjecture about the difference between the first two Dirichlet eigenvalues (the fundamental gap) of any convex domain in the Euclidean plane. Recently, progress has been made in the case where the domains are polygons and, in particular, triangles. We examine the conjecture for triangles in hyperbolic geometry, though we seek an for an upper bound for the fundamental gap rather than a lower bound.

Invariant barriers to reactive front propagation in fluid flows

We present theory and experiments on the dynamics of reaction fronts in two-dimensional, vortex-dominated flows, for both time-independent and periodically driven cases. We find that the front propagation process is controlled by one-sided barriers that are either fixed in the laboratory frame (time-independent flows) or oscillate periodically (periodically driven flows). We call these barriers burning invariant manifolds (BIMs), since their role in front propagation is analogous to that of invariant manifolds in the transport and mixing of passive impurities under advection. Theoretically, the BIMs emerge from a dynamical systems approach when the advection-reaction-diffusion dynamics is recast as an ODE for front element dynamics. Experimentally, we measure the location of BIMs for several laboratory flows and confirm their role as barriers to front propagation.

Barriers to front propagation in ordered and disordered vortex flows

We present experiments on reactive front propagation in a two-dimensional (2D) vortex chain flow (both time-independent and time-periodic) and a 2D spatially disordered (time-independent) vortex-dominated flow. The flows are generated using magnetohydrodynamic forcing techniques, and the fronts are produced using the excitable, ferroin-catalyzed Belousov-Zhabotinsky chemical reaction. In both of these flows, front propagation is dominated by the presence of burning invariant manifolds (BIMs) that act as barriers, similar to invariant manifolds that dominate the transport of passive impurities. Convergence of the fronts onto these BIMs is shown experimentally for all of the flows studied. The BIMs are also shown to collapse onto the invariant manifolds for passive transport in the limit of large flow velocities. For the disordered flow, the measured BIMs are compared to those predicted using a measured velocity field and a three-dimensional set of ordinary differential equations that describe the dynamics of front propagation in advection-reaction-diffusion systems.

Geophysical Investigation of Subsurface Three-Dimensional Architecture of Icy Debris Fans With Ground Penetrating Radar in the Wrangell Mountains, Alaska

Icy debris fans have are newly-described landforms (Kochel and Trop, 2008 and 2012) as landforms developed immediately after deglaciation on Earth and similar features have been observed on Mars. Subsurface characteristics of Icy debris fans have not been previously investigated. Ground penetrating radar (GPR) was used to non-invasively investigate the subsurface characteristics of icy debris fans near McCarthy, Alaska, USA. The three fans investigated in Alaska are the East, West, and Middle fans (Kochel and Trop, 2008 and 2012) which below the Nabesna ice cap and on top of the McCarthy Creek Glacier. Icy debris fans in general are a largely unexplored suite of paraglacial landforms and processes in alpine regions. Recent field studies focused on direct observations and depositional processes. Their results showed that the fan's composition is primarily influenced by the type and frequency of depositional processes that supply the fan. Photographic studies show that the East Fan receives far more ice and snow avalanches whereas the Middle and West Fans receive fewer mass wasting events but more clastic debris is deposited on the Middle and West fan from rock falls and icy debris flows. GPR profiles and Wide-angle reflection and refraction (WARR) surveys consisting of both, common mid-point (CMP), and common shot-point (CSP) surveys investigated the subsurface geometry of the fans and the McCarthy Creek Glacier. All GPR surveys were collected in July of 2013 with 100MHz bi-static antennas. Four axial profiles and three cross-fan profiles were done on the West and Middle fans as well as the McCarthy Creek Glacier in order to investigate the relationship between the three features. GPR profiles yielded reflectors that were continuous for 10+ m and hyperbolic reflections in the subsurface. The depth to these reflections in the subsurface requires knowledge of the velocity of the subsurface. To find the velocity of the subsurface eight WARR surveys collected on the fans and on the McCarthy Creek glacier to provide information on variability of subsurface velocities. The profiles of the Middle and West fan have more reflections in their profiles compared to profiles done on the McCarthy Creek Glacier. Based on the WARR surveys, we interpret the lower energy return in the glacier to be caused by two reasons. 1) The increased attenuation due to wet ice versus drier ice and on the fan with GPR velocities >0.15m/ns. 2) Lack of interfaces in the glacier compared to those in the fans which are inferred to be produced by the alternating layers of stratified ice and lithic-rich layers. The GPR profiles on the West and Middle Fans show the shallow subsurface being dominated by lenticular reflections interpreted to be consistent with the shape of surficial deposits. The West Fan is distinguished from the Middle Fan by the nature of its reflections patterns and thicknesses of reflection packages that clearly shows the Middle fan with a greater thickness. The changes in subsurface reflections between the Middle and West Fans as well as the McCarthy Creek Glacier are thought to reflect the type and frequency of depositional processes and surrounding bedrock and talus slopes.