  Invariant parameterization schemes for the eddy-vorticity flux in the
barotropic vorticity equation on the beta-plane are constructed and then
applied to turbulence modeling. This construction is realized by the exhaustive
description of differential invariants for the maximal Lie invariance
pseudogroup of this equation using the method of moving frames, which includes
finding functional bases of differential invariants of arbitrary order, a
minimal generating set of differential invariants and a basis of operators of
invariant differentiation in an explicit form. 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 shape of energy spectrum
in the enstrophy inertial range. By presenting conservative invariant
hyperdiffusion terms, we also demonstrate that the concepts of invariant and
conservative parameterizations are consistent.
