In order to understand the fundamental parameters governing glacier advance and retreat, and also the spectral properties of fluctuations in glacier length in response to noisy weather, we examine outputs of a numerical flowline model solving the shallow-ice equations with sliding. The numerical results reveal a surprising simplicity: the time evolution and spectral shape of glacier excursions depend on a single parameter, a time constant determined by the geometrical properties of the glacier. Furthermore, the numerical results reveal that perturbations in mass balance over the glacier surface set in motion a sequence of events that can be roughly described as occurring in three overlapping stages: (1) changes in interior thickness drive (2) changes in terminus flux, which in turn drive (3) changes in glacier length. A simple, third-order linear differential equation, which extends previous models in the literature, successfully captures these important features of the glacier flow. This three-stage linear model is readily invertible to recover climate history. It provides clear physical insight and analytical expressions for some important metrics of glacier behavior, such as variance, sensitivity and excursion probabilities. Finally, it facilitates uncertainty analysis. The linear model can also be adapted for arbitrary catchment geometry, and is applied to Nigardsbreen, Norway.