For glaciological conditions typical of valley glaciers, the central idea of this research lies in understanding the effects of high-order mechanics and parameterizing these for simpler dynamical and statistical methods in glaciology. As an effective tool for this, I formulate a new brand of dynamical models that describes distinct physical processes of deformational flow. Through numerical simulations of idealized glacier domains, I calculate empirical correction factors to capture the effects of longitudinal stress gradients and lateral drag for simplified dynamical models in the plane-strain regime. To get some insights into real glacier dynamics, I simulate Haig Glacier in the Canadian Rocky Mountains. As geometric effects overshadow dynamical effects in glacier retreat scenarios, it appears that high-order physics are not very important for Haig Glacier, particularly for evaluating its fate. Indeed, high-order and reduced models all predict that Haig Glacier ceases to exist by about AD 2080 under ongoing climate warming. This finding regarding the minimal role of high-order physics may not be broadly valid, as it is not true in advance scenarios at Haig Glacier and it may not be representative of other glaciological settings.
Through a `bulk' parameterization of high-order physics, geometric and climatic settings, sliding conditions, and transient effects, I also provide new insights into the volume-area relation, a widely used statistical method for estimating glacier volume. I find a steady-state power-law exponent of 1.46, which declines systematically to 1.38 after 100 years of sustained retreat, in good accord with the observations. I recommend more accurate scaling relations through characterization of individual glacier morphology and degree of climatic disequilibrium. This motivates a revision of global glacier volume estimates, of some urgency in sea level rise assessments.