Symmetric Truncations of the Shallow Water Equations

Abstract

Abstract : Conservation of potential vorticity in Eulerian fluids reflects particle interchange symmetry in the Lagrangian fluid version of the same theory. The algebra associated with this symmetry in the shallow water equations is studied here, and we give a method for truncating the degrees of freedom of the theory which preserves a maximal number of invariants associated with this algebra. The symmetry associated with keeping N modes of the shallow water flow is SU(N). In the limit where the number of modes goes to infinity (N approaches limit of infinity), all the conservation laws connected with potential vorticity conservation are recovered. We also present a Hamiltonian which is invariant under this truncated symmetry and which reduces to the familiar shallow water Hamiltonian when N approaches limit of infinity. All this provides a finite dimensional framework for numerical work with the shallow water equations which preserves not only energy and enstrophy but all other known conserved quantities consistent with the finite number of degrees of freedom. The extension of these ideas to other nearly two dimensional flow is discussed.