Symmetric truncations of the shallow-water equations.

Conservation of potential vorticity in Eulerian uids re ects particle interchange symmetry in the Lagrangian uid 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 nite dimensional symmetry associated with keeping only N modes of the shallow water ow is SU(N). In the limit where the number of modes goes to in nity (N ! 1), 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 !1. All this provides a nite 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 nite number of degrees of freedom. The extension of these ideas to other nearly two dimensional ows is discussed. PACS numbers: 47.10.+g, 47.15.Ki, 92.90.+x, 02.20.Tw 1