describes an ocean in hydrostatic and geostrophic balance with water of constant density, bounded by rigid upper A spectral numerical scheme is developed for simulations of twodimensional incompressible fluid flow in a circular basin. The vorticand lower boundaries. It can be shown [9] that this system ity and streamfunction fields are represented by products of Jacobi is governed by the horizontal advection of absolute vorticpolynomials and complex exponentials. The Jacobi polynomials ity q with a nondivergent velocity field v ϭ k ϫ ٌ, where are used for the radial dependence of the fields, and the complex k is a unit vector pointing vertically upward and is a exponentials for the angular dependence. The basis functions are orthogonal with respect to the natural inner product for a circular streamfunction. The basin was taken to be circular, with domain. It is demonstrated how the Laplace operator and its inverse zero at the boundary (free-slip), and the flow was ascan be expressed exactly in terms of these basis functions. It is also sumed to be forced by a temporally constant and spatially shown how the advection term can be evaluated without aliasing, uniform input of vorticity and damped by Ekman friction.
making use of a transform grid with equidistant angular values and
The equation studied was the time-independent version of 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 con-Ѩq Ѩt ϩ J(, q) ϩ ϩ ϭ 0.
(1)
servation 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. 115-131 Here q is the absolute vorticity, q ϭ ϩ f, where the relative 136). ᮊ 1997 Academic Press vorticity ϭ k • ٌ ϫ v ϭ ٌ 2 and f is the planetary vorticity. The operators ٌ 2 and J are the Laplace and Jacobi operators, respectively. Use of the latter operator is common prac-100