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This work analyses the homogeneous isotropic turbulence by means of theequivalence between Euler and Lagrange representations of motion, adopting thebifurcation rates associated with Navier--Stokes and kinematic equations, andan appropriate hypothesis of fully developed chaos. The equivalence of thesemotion descriptions allows to show that kinetic and thermal energy cascadearise both from the convective term of Liouville equation. Accordingly, thesephenomena, of nondiffusive nature, correspond to a transport in physical spacelinked to the trajectories divergence. Both the bifurcation rates are properlydefined, where the kinematic bifurcation rate is shown to be much greater thanNavier--Stokes bifurcation rate. This justifies the proposed hypothesis offully developed chaos where velocity field and particles trajectoriesfluctuations are statistically uncorrelated. Thereafter, a specific ergodicproperty is presented, which relates the statistics of fluid displacement tothat of velocity and temperature fields. A detailed analysis of separation rateis proposed which studies the statistics of radial velocity component along thematerial separation vector. Based on previous elements, the closure formulas ofvon K\'arm\'an--Howarth and Corrsin equations are finally achieved. Theseclosures, of nondiffusive kind, represent a propagation phenomenon, andcoincide with those just presented by the author in previous works,corroborating the results of these latter.

Turbulent energy cascade through equivalence of Euler and Lagrange motion descriptions and bifurcation rates