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UBC Theses and Dissertations

Accurate modelling of glacier flow Waddington, Edwin Donald


Recent interest in climatic change and ice .sheet variations points out the need for accurate and numerically stable models of time-dependent ice masses. Little attention has been paid to this topic by the glaciological community, and there is good reason to believe that much of the published literature on numerical modelling of the flow of glaciers and ice sheets is quantitatively incorrect. In particular, the importance of the nonlinear instability has not been widely recognized. The purposes of this thesis are to develop and to verify a new numerical model for glacier flow, compare the model to another widely accepted model, and to demonstrate the model in several glaciologically interesting applications. As in earlier work, the computer model solves the continuity equation together with a flow law for ice. Thickness profiles along flow lines are obtained as a function of time for a temperate ice mass with arbitrary bed topography and mass balance. A set of necessary tests to be satisfied by any numerical model of glacier flow is presented. The numerical solutions are compared with analytical solutions; these include a simple thickness-velocity relation to check terminus mobility, and Burgers* equation to check continuity and dynamic behaviour with full nonlinearity. An attempt has been made to verify the accuracy of the computer model of Budd and Mclnnes (1974), Rudd (1975) and Mclnnes (unpublished). These authors have reported problems with numerical instability. If the existing documentation is accurate, the Budd-Mclnnes model appears to suffer from mass conservation violations both locally and globally. The new numerical model developed in this thesis can be used to reconstruct the velocity field within the glacier at each time step; this velocity field satisfies continuity and Glen's flow law for ice. Integration of this velocity field yields the trajectories of individual ice elements flowing through the time-varying ice mass. The trajectories and velocity field are checked by comparison with an analytical solution for a steady state ice sheet (Nagata, 1977). The model in this thesis is not restricted to steady state, and it avoids the violations of mass conservation, and the approximations about the velocity field found in some published trajectory models. The feasibility of using stable isotopes to investigate prehistoric surging of valley glaciers has been studied with a model simulating the Steele Glacier, Yukon Territory. A sliding < velocity and surge duration were specified, based on the observations of the 1966-67 surge. A surge period of roughly 100 years gave the most realistic ice thickness throughout the surge cycle. By calculating ice trajectories and using two plausible relationships between 6(01B/016) and position or height, longitudinal sections and surface profiles of 6 were constructed for times before, during, and after a surge. Discontinuities of up to 0.8°/Oo were found across several surfaces dipping upstream into the glacier. Each of these surfaces is the present location of the ice which formed the ice-air interface at the time a previous surge began. It may be difficult to observe these surfaces on the Steele Glacier due to the large and poorly-understood background variability of 6. The generation of wave ogives has been examined following the theory of Nye (I958[b])r wherein waves are caused by a combination of seasonal variation in mass balance and plastic deformation in an icefall. The wave train generated on a glacier is shown in this thesis to be a convolution of the velocity gradient with an integral of the mass balance function. This integral is the impulse response of the glacier surface to a step in the velocity function. Spatial variations in the glacier width and mass balance also contribute to the wave train. This formulation is used to explain why many icefalls do not generate wave ogives in spite of large seasonal balance variations and large plastic deformations.

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