On Contour Crossings in Contour-Advective Simulations of Geophysical Fluid Flows

Contour crossings are a form of numerical error common to contour dynamical and contour-advective simulations of geophysical fluid flows. This thesis is devoted to the study of contour crossings that occur in simulations of fluid motion performed by the Contour-Advective Semi-Lagrangian (CASL) algorithm. Components of the CASL algorithm relevant to the study of contour crossings are discussed. Of particular importance are the CASL algorithm’s methods for discretely representing contours, redistributing nodes along contours and enforcing a minimum scale on simulations via contour surgery. With knowledge of each of these components of the CASL algorithm it is possible to examine various techniques for preventing contour crossings. Two techniques for reducing error due to contour crossings are discussed in this thesis. The first of these techniques involves the insertion of nodes along contours at points of close approach to other contours according to a set of geometric conditions similar to those used by fusion surgery. This technique was initially described by P. W. C. Vosbeek and R. M. M. Mattheij, J. Comput. Phys., 133, pp 222-234, 1997. The second technique is simpler in both concept and implementation; it involves multiplying the standard CASL algorithm’s node density by a constant. A part analytic, part numerical method for estimating the effectiveness of node density multiplication relative to node insertion following Vosbeek and Mattheij is described. Associated with the study of contour crossings is the importance of developing a method for quantifying numerical error due to their presence. The method described in this thesis treats each contour as a simple polygon. Any points of intersection between contours are detected and the relative severity of each crossing is determined by measuring an area in error. This area may either be an area of intersection for initially separated contours or an area of extrusion for initially nested contours. A comprehensive study of contour crossing errors that occur in CASL simulations of two dimensional turbulence in a cylinder is made. The effects of varying spatial resolution of contours by increasing node density, inserting nodes in regions of close interaction between contours or altering the resolution of the velocity inversion grid are observed. Crossing errors are quantified in terms of the number of contour crossings, c, the total area in error, at, and the mean area of individual regions in error, am. An estimate for the error in total enstrophy due to contour crossings is obtained. It is found that the error due to contour crossings in turbulence simulations is small,