Understanding understanding

The poster presents research into conceptualising understanding in physics education

Tameness on the boundary and Ahlfors' measure conjecture

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N has non-empty conformal boundary, (2) N is not homotopy equivalent to a compression body, or (3) N is a strong limit of geometrically finite manifolds. The first case proves Ahlfors' measure conjecture for Kleinian groups in the closure of the geometrically finite locus: given any algebraic limit G of geometrically finite Kleinian groups, the limit set of G is either of Lebesgue measure zero or all of the Riemann sphere. Thus, Ahlfors' conjecture is reduced to the density conjecture of Bers, Sullivan, and Thurston.Comment: New revised version, 22 pages. To appear, Publ. I.H.E.S. This version represents a fairly substantial reorganization of the logical structure of the pape

Development of roll waves in open channels

This study is concerned with some of the properties of roll waves that develop naturally from a turbulent uniform flow in a wide rectangular channel on a constant steep slope. The wave properties considered were depth at the wave crest, depth at the wave trough, wave period, and wave velocity. The primary focus was on the mean values and standard deviations of the crest depths and wave periods at a given station and how these quantities varied with distance along the channel. The wave properties were measured in a laboratory channel in which roll waves developed naturally from a uniform flow. The Froude number F (F = u_n/√gh_n, u_n = normal velocity, h_n = normal depth, g = acceleration of gravity) ranged from 3.4 to 6.0 for channel slopes S_0 of .05 and . 12 respectively. In the initial phase of their development the roll waves appeared as small amplitude waves with a continuous water surface profile. These small amplitude waves subsequently developed into large amplitude shock waves. Shock waves were found to overtake and combine with other shock waves with the result that the crest depth of the combined wave was larger than the crest depths before the overtake. Once roll waves began to develop, the mean value of the crest depths h_(max) increased with distance. Once the shock waves began to overtake, the mean wave period T_(av) increased approximately linearly with distance. For a given Froude number and channel slope the observed quantities h_(max)/h_n, T' (T' = S_0 T_(av) √g/h_n), and the standard deviations of h_(max)/h_n and T', could be expressed as unique functions of ℓ /h_n (ℓ= distance from beginning of channel) for the two-fold change in h_n occurring in the observed flows. A given value of h_(max)h_n occurred at smaller values of ℓ/h_n as the Froude number was increased. For a given value of h_(max) /h_n the growth rate ∂h_(max)/∂ℓ of the shock waves increased as the Froude number was increased. A laboratory channel was also used to measure the wave properties of periodic permanent roll waves. For a given Froude number and channel slope the h_(max)/h_n vs. T' relation did not agree with a theory in which the weight of the shock front was neglected. After the theory was modified to include this weight, the observed values of h_(max)/h_n were within an average of 6.5 percent of the predicted values, and the maximum discrepancy was 13.5 percent. For h_(max)/h_n sufficiently large (h_(max)/h_n > approximately 1.5) it was found that the h_(max)/h_n vs. T' relation for natural roll waves was practically identical to the h_(max)/h_n vs. T' relation for periodic permanent roll waves at the same Froude number and slope. As a result of this correspondence between periodic and natural roll waves, the growth rate ∂h_(max)/∂ℓ of shock waves was predicted to depend on the channel slope, and this slope dependence was observed in the experiments

Representing moisture fluxes and phase changes in glacier debris cover using a reservoir approach

Due to the complexity of treating moisture in supraglacial debris, surface energy balance models to date have neglected moisture infiltration and phase changes in the debris layer. The latent heat flux (QL) is also often excluded due to the uncertainty in determining the surface vapour pressure. To quantify the importance of moisture on the surface energy and climatic mass balance (CMB) of debris-covered glaciers, we developed a simple reservoir parameterization for the debris ice and water content, as well as an estimation of the latent heat flux. The parameterization was incorporated into a CMB model adapted for debris-covered glaciers. We present the results of two point simulations, using both our new “moist” and the conventional “dry” approaches, on the Miage Glacier, Italy, during summer 2008 and fall 2011. The former year coincides with available in situ glaciological and meteorological measurements, including the first eddy-covariance measurements of the turbulent fluxes over supraglacial debris, while the latter contains two refreeze events that permit evaluation of the influence of phase changes. The simulations demonstrate a clear influence of moisture on the glacier energy and mass-balance dynamics. When water and ice are considered, heat transmission to the underlying glacier ice is lower, as the effective thermal diffusivity of the saturated debris layers is reduced by increases in both the density and the specific heat capacity of the layers. In combination with surface heat extraction by QL, subdebris ice melt is reduced by 3.1% in 2008 and by 7.0% in 2011 when moisture effects are included. However, the influence of the parameterization on the total accumulated mass balance varies seasonally. In summer 2008, mass loss due to surface vapour fluxes more than compensates for the reduction in ice melt, such that the total ablation increases by 4.0 %. Conversely, in fall 2011, the modulation of basal debris temperature by debris ice results in a decrease in total ablation of 2.1 %. Although the parameterization is a simplified representation of the moist physics of glacier debris, it is a novel attempt at including moisture in a numerical model of debris-covered glaciers and one that opens up additional avenues for future research