Isotropic Navier-Stokes turbulence. I. Qualitative features and basic equations

This set of two lectures is intended to give a taste of modern theoretical approaches to isotropic turbulence in an incompressible Navier-Stokes (NS) fluid. The first lecture is devoted to basic mathematical properties of the NS equation and qualitative physical features of turbulence. The second lecture surveys some theoretical approaches and then outlines the use of stochastic modeling to obtain self-consistent equations for low-order moments of the velocity field. Equation numbering reflects the lecture number, the section number, and then the consecutive equation number. Fully developed NS turbulence appears to exhibit an irreducibly large number of degrees of freedom (1018 is not unusual in the atmosphere). It has not yielded to the techniques that have been powerful for small dynamical systems. Instead, it has pointed to the need for statistical approaches that can handle the strongly nonlinear interaction of huge numbers of modes that are far from absolute statistical equilibrium. 17 refs