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Kadakia, N, Rey D, Ye J, Abarbanel HDI.  2017.  Symplectic structure of statistical variational data assimilation. Quarterly Journal of the Royal Meteorological Society. 143:756-771.   10.1002/qj.2962   Abstract

Data assimilation variational principles (4D-Var) exhibit a natural symplectic structure among the state variables x(t) and. x(t). We explore the implications of this structure in both Lagrangian coordinates {x(t), x(t)} andHamiltonian canonical coordinates {x(t), p(t)} through a numerical examination of the chaotic Lorenz 1996 model in ten dimensions. We find that there are a number of subtleties associated with discretization, boundary conditions, and symplecticity, suggesting differing approaches when working in the the Lagrangian versus the Hamiltonian description. We investigate these differences in detail, and accordingly develop a protocol for searching for optimal trajectories in a Hamiltonian space. We find that casting the problem into canonical coordinates can, in some situations, considerably improve the quality of predictions.

Rouhi, A, Abarbanel HDI.  1993.  Symmetrical truncations of the shallow-water equations. Physical Review E. 48:3643-3655.   10.1103/PhysRevE.48.3643   AbstractWebsite

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 finite-dimensional symmetry associated with keeping only N modes of the shallow-water flow is SU(N). In the limit where the number of modes goes to infinity (N --> 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 --> 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 flows is discussed.