Invariant parameterization schemes for turbulence modeling − 2021
Bihlo Alex1, Cardoso-Bihlo Elsa1 and Popovych Roman2
(1Memorial University of Newfoundland, Canada;
2 University of Vienna, Austria and Institute of Mathematics, Kyiv)
Invariant parameterization schemes for turbulence modeling
Abstract:
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 $k^{-3}$ shape of the 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.
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