Lagrangian tetrad dynamics and the phenomenology of turbulence

A new phenomenological model of turbulent fluctuations is constructed by co nsidering the Lagrangian dynamics of four points (the tetrad), The closure of the equations of motion is achieved by postulating an anisotropic, i.e., tetrad shape dependent, relation of the local pressure and the velocity gr adient defined on the tetrad. The nonlocal contribution to the pressure and the incoherent small scale fluctuations are modeled as Gaussian white "noi se." The resulting stochastic model for the coarse-grained velocity gradien t is analyzed approximately, yielding predictions for the probability distr ibution functions of different second- and third-order invariants, The resu lts are compared with the direct numerical simulation of the Navier-Stokes. The model provides a reasonable representation of the nonlinear dynamics i nvolved in energy transfer and vortex stretching and allows the study of in teresting aspects of the statistical geometry of turbulence, e.g., vorticit y/strain alignment, In a state with a constant energy flux (and K41 power s pectrum), it exhibits the anomalous scaling of high moments associated with formation of high gradient sheets-events associated with large energy tran sfer. An approach to the more complete analysis of the stochastic model, pr operly including the effect of fluctuations, is outlined and will enable fu rther quantitative juxtaposition of the model with the results of the direc t numerical simulations. (C) 1999 American Institute of Physics, [S1070-663 1(99)02708-7].