We study shell models that conserve the analogues of energy and enstrophy,
hence designed to mimic fluid turbulence in 2D. The main result is that the
observed state is well described as a formal statistical equilibrium, closely
analogous to the approach to two-dimensional ideal hydrodynamics of Onsager,
Hopf and Lee. In the presence of forcing and dissipation we observe a forward
flux of enstrophy and a backward flux of energy. These fluxes can be understood
as mean diffusive drifts from a source to two sinks in a system which is close
to local equilibrium with Lagrange multipliers (``shell temperatures'')
changing slowly with scale. The dimensional predictions on the power spectra
from a supposed forward cascade of enstrophy, and from one branch of the formal
statistical equilibrium, coincide in these shell models at difference to the
corresponding predictions for the Navier-Stokes and Euler equations in 2D. This
coincidence have previously led to the mistaken conclusion that shell models
exhibit a forward cascade of enstrophy.Comment: 25 pages + 9 figures, TeX dialect: RevTeX 3.
