Hidden conservation laws in hydrodynamics; energy and dissipation rate fluctuation spectra in strong turbulence
Abstract
The Hamiltonian formulation of hydrodynamics in Clebsch variables is used for construction of a statistical theory of turbulence. The statistics of the Clebsch field is assumed to be close to Gaussian. It is shown that the energy spectrum consists of two ranges with E(k)≈k ^{{5}/{3}} and E( k)≈ k^{1}. The spectrum of the dissipation rate fluctuations has three scaling regimes: E^{ɛ}( k)≈ k^{1}; k ^{{1}/{3}} and k^{0} at the large, intermediate and small scales, respectively. The origin of the exponential distribution of velocity differences is discussed. The new scaling regime corresponds to a hidden conservation law, discovered in the Clebsch formulation of hydrodynamics. It is shown that viscous effects are responsible for production of the conserved quantity. The theoretical predictions are compared with results of numerical simulations of decaying turbulence.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 April 1993
 DOI:
 10.1016/01672789(93)90050B
 Bibcode:
 1993PhyD...64..379Y