Poisson-Bracket Approach to the Construction of Energy- and Potential-Enstrophy-Conserving Algorithms for the Shallow-Water Equations

Abstract

Arakawa and Lamb discovered a finite-difference approximation to the shallow-water equations that exactly conserves finite-difference approximations to the energy and potential enstrophy of the fluid. The Arakawa? Lamb (AL) algorithm is a stunning and important achievement?stunning, because in the shallow-water case, neither energy nor potential enstrophy is a simple quadratic, and important because the simultaneous conservation of energy and potential enstrophy is known to prevent the spurious cascade of energy to high wavenumbers. However, the method followed by AL is somewhat ad hoc, and it is difficult to see how it might be generalized to other systems. In this paper, the AL algorithm is rederived and greatly generalized in a way that should permit still further generalizations. Beginning with the Hamiltonian formulation of shallow-water dynamics, its two essential ingredients?the Hamiltonian functional and the Poisson-bracket operator?are replaced by finite-difference approximations that maintain the desired conservation laws. Energy conservation is maintained if the discrete Poisson bracket retains the antisymmetry property of the exact bracket, a trivial constraint. Potential enstrophy is conserved if a set of otherwise arbitrary coefficients is chosen in such a way that a very large quadratic form contains only diagonal terms. Using a symbolic manipulation program to satisfy the potential-enstrophy constraint, it is found that the energy- and potential-enstrophy-conserving schemes corresponding to a stencil of 25 grid points contain 22 free parameters. The AL scheme corresponds to the vanishing of all free parameters. No parameter setting can increase the overall accuracy of the schemes beyond second order, but 19 of the free parameters may be independently adjusted to yield a scheme with fourth-order accuracy in the vorticity equation.