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Unformatted text preview: INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY Nonlinearity 18 (2005) R1–R16 doi:10.1088/09517715/18/5/R01 INVITED ARTICLE A general method for conserving quantities related to potential vorticity in numerical models Rick Salmon Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 920930213, USA Email: [email protected] Received 14 March 2005, in final form 10 May 2005 Published 6 June 2005 Online at stacks.iop.org/Non/18/R1 Recommended by K Ohkitani Abstract Nambu proposed a generalization of Hamiltonian dynamics in the form d F/ d t = { F, H, Z } , which conserves H and Z because the Nambu bracket { F, H, Z } is completely antisymmetric. The equations of fluid dynamics fit Nambu’s form with H the energy and Z a quantity related to potential vorticity. This formulation makes it easy, in principle, to construct numerical fluid models that conserve analogues of H and Z ; one need only discretize the Nambu bracket in such a way that the antisymmetry property is preserved. In practice, the bracket may contain apparent singularities that are cancelled by the functional derivatives of Z . Then the discretization must be carried out in such a way that the cancellation is maintained. Following this strategy, we derive numerical models of the shallowwater equations and the equations for incompressible flow in two and three dimensions. The models conserve the energy and an arbitrary moment of the potential vorticity. The conservation of potential enstrophy—the second moment of potential vorticity—is thought to be especially important because it prevents the spurious cascade of energy into high wavenumbers. Mathematics Subject Classification: 65P10 1. Introduction The equations of fluid dynamics fit the Hamiltonian form d F d t = { F, H } , (1.1) where F is an arbitrary functional of the fields representing the state of the fluid; H is the Hamiltonian functional; and { , } is the Poisson bracket, an antisymmetric, bilinear 09517715/05/050001+16$30.00 © 2005 IOP Publishing Ltd and London Mathematical Society Printed in the UK R1 R2 Invited Article operator that obeys the Jacobi identity ( 5.3 ). For example, the equations for twodimensional, incompressible flow may be written in the form ∂ζ ∂t = J (ζ, ψ), (1.2) where ζ = ∇ 2 ψ (1.3) is the vorticity of the fluid; ψ(x, y, t) is the stream function; (u, v) = ( − ψ y , ψ x ) is the velocity in the (x, y) direction; and J (A, B) ≡ ∂(A, B) ∂(x, y) (1.4) is the Jacobian operator in two dimensions. For simplicity, we consider only periodic boundary conditions. The dynamics ( 1.2 ) and ( 1.3 ) fits the form ( 1.1 ) with { F, H } ≡ ZZ d x ζ J (F ζ , H ζ ), (1.5) where F ζ ≡ δF/δζ denotes the functional derivative, H [ ζ(x, y) ] = 1 2 ZZ d x ∇ ψ · ∇ ψ (1.6) and ψ and ζ are related by ( 1.3 ) and the periodic boundary conditions. We note that δH/δζ = − ψ and δH/δψ = − ζ ....
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 Spring '08
 Iskandarani,M
 Numerical Analysis, Hamiltonian mechanics, Arakawa, vorticity, shallowwater Casimir, Casimirs

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