  We suggest a new focus for turbulence studies -- multi-mode correlations --
which reveal the hitherto hidden nature of turbulent state. We apply this
approach to shell models describing basic properties of turbulence. The family
of such models allows one to study turbulence close to thermal equilibrium,
which happens when the interaction time weakly depends on the mode number. As
the number of modes increases, the one-mode statistics approaches Gaussian
(like in weak turbulence), the occupation numbers grow, while the three-mode
cumulant describing the energy flux stays constant. Yet we find that higher
multi-mode cumulants grow with the order. We derive analytically and confirm
numerically the scaling law of such growth. The sum of all squared
dimensionless cumulants is equal to the relative entropy between the full
multi-mode distribution and the Gaussian approximation of independent modes; we
argue that the relative entropy could grow as the logarithm of the number of
modes, similar to the %mutual information and entanglement entropy in critical
phenomena. Therefore, the multi-mode correlations give a new way to
characterize turbulence states and possibly divide them into universality
classes.
