Flow-oriented Discretization Using the Land Surface Stream Function

Abstract:

A natural discretisation for modelling surface water flow is the flow-oriented mesh based on contour lines and slope lines. This mesh allows the modelling of the two-dimensional flow field as a series of one-dimensional flows, with the mesh itself capturing the divergence and convergence of flow due to topography.

Algorithms for creating flow-oriented meshes originated in the 1980s (O'Loughlin, 1986; Moore et al, 1988; Dawes and Short, 1994; Maunder, 1999). These meshes were all derived from contour data and the processes were invariably error prone and incomplete until the method of Moretti and Orlandini (2008) which provided a definitive solution. However, contours are not an ideal data source because they lack information on surface form between contours and are sparse in valley floors where flow concentrates.

An alternative approach is to build the mesh elements from rasters of elevation and stream function. The land surface stream function is a field of values, complementary to elevation, that assigns a unique value to each slope line. The difference in value between two adjacent lines is the horizontal projected area between those lines. The stream function (psi) and elevation (z) create a curvilinear coordinate system and flow-oriented elements are rectangles in the (psi,z) space.

Deriving the stream function from a grid DEM is non-trivial. Stream function is mostly continuous but has extremely steep spatial gradients at valley and ridge lines and discontinuities along some ridge and valley lines, requiring special measures to ensure a consistent representation.

Once the stream function has been computed, mesh elements can be constructed by first identifying whole hillslopes and then subdividing along contours and slope lines until the elements are small enough to represent the spatial patterns of interest. The stream function resolves all the complexities of the topological structure of flow lines that bedevil the contour-based methods, including handling of saddles and depressions and representing the hydrological significance of ridges and valleys.