A Spectral Model for Two-Dimensional Incompressible Fluid Flow in a Circular Basin

Abstract

A spectral numerical scheme is developed for simulations of two-dimensional incompressible fluid flow in a circular basin. The vorticity and streamfunction fields are represented by products of Jacobi polynomials and complex exponentials. The Jacobi polynomials are used for the radial dependence of the fields, and the complex exponentials for the angular dependence. The basis functions are orthogonal with respect to the natural inner product for a circular domain. It is demonstrated how the Laplace operator and its inverse can be expressed exactly in terms of these basis functions. It is also shown how the advection term can be evaluated without aliasing, making use of a transform grid with equidistant angular values and Gaussian radial values. It is shown that without forcing and friction the model conserves absolute enstrophy and circulation and, if the planetary vorticity is circularly symmetric, also angular momentum. The model does not conserve energy. However, the degree of conservation of energy rapidly increases with increasing resolution. Examples of time integrations will be discussed in the companion paper (Part II; W. T. M. Verkley, 1997,J. Comput. Phys.115–131136).