Physics – Mathematical Physics
Scientific paper
2011-12-08
Physics
Mathematical Physics
25 pages, 4 figures
Scientific paper
Differential invariants are exhaustively described for the maximal Lie invariance pseudogroup of the barotropic vorticity equation on the beta-plane. Functional bases of differential invariants of arbitrary order, a minimal generating set of differential invariants and a basis of operators of invariant differentiation are found for this Lie pseudogroup in an explicit form and applied to construct invariant parameterization schemes for the eddy-vorticity flux in the beta-plane equation. Special attention is paid to the problem of two-dimensional turbulence on the beta-plane. It is shown that classical hyperdiffusion as used to initiate the energy-enstrophy cascades violates the symmetries of the vorticity equation. Invariant but nonlinear hyperdiffusion-like terms of new types are introduced and then used in the course of numerically integrating the vorticity equation and carrying out freely decaying turbulence tests. It is found that the invariant hyperdiffusion scheme is close to but not exactly reproducing the 1/k^3 shape of energy spectrum in the enstrophy inertial range. Conservative invariant hyperdiffusion terms are also constructed.
Bihlo Alexander
Popovych Roman O.
Santos Cardoso-Bihlo Elsa Dos
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