It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh P'{e}clet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, are then used to characterise the oscillations of the discrete solution in terms of the mesh P'{e}clet number and boundary conditions of the problem. (Also cross-referenced UMIACS-TR-2000-15)