  The framework of invariant parameterization is extended to higher-order
closure schemes. We also define, for the first time, generalized invariant
parameterization schemes, where symmetries of the corresponding original model
are preserved as equivalence transformations of related classes of closed
system of differential equations. As a particular problem, we consider
invariant parameterization schemes for geostrophic eddies in a barotropic
ocean. Here the initial model is the barotropic vorticity equation, which is
equivalent to the system of incompressible inviscid two-dimensional Euler
equations on a midlatitude beta-plane. The maximal Lie invariance algebra of
this model is infinite-dimensional, and we intend to preserve it in the course
of invariant parameterization, at least partially. The parameterizations
proposed for the eddy vorticity flux and the energy flux are of order one and a
half since we explicitly consider the equation for the turbulent kinetic energy
in the closure models. These parameterizations are therefore the first examples
of invariant higher-order closure schemes. Numerical experiments are carried
out to assess the performance of these invariant schemes in studies of freely
decaying two-dimensional turbulence, and it is verified that the invariant
parameterization schemes give, on average, better results than the standard
non-invariant closure models do.
